We survey ultradifferentiable extension theorems, i.e., quantitative versions of Whitney's classical extension theorem,
with special emphasis on the existence of continuous linear extension operators.
The focus is on Denjoy–Carleman classes for which we develop the theory from scratch and discuss important related concepts
such as (non-)quasianalyticity.
It allows us to give an efficient and, to a fair extent, elementary
introduction to Braun–Meise–Taylor classes based on their representation
as intersections and unions of  Denjoy–Carleman classes.

The author was supported by the Austrian Science Fund (FWF), Grant P 32905-N and START Programme Y963

[2020]

§ INTRODUCTION

The development of differential analysis in the last century was strongly influenced by Whitney's work on differentiable functions.
Especially fruitful was Whitney's extension theorem which describes the restrictions of smooth functions on $\R^n$ to closed subsets.
For a closed non-empty set $A \subseteq \R^n$ we consider the restriction map

\[
j^\infty_A : C^\infty(\R^n) \to C^0(A)^{\N^n}, \quad f \mapsto (f^{(\al)}|_A)_{\al \in \N^n}.
\]

Taylor's theorem implies that the image $j^\infty_A  C^\infty(\R^n)$ is contained in the set $\cE(A)$ of Whitney jets:
$F= (F^\al)_\al \in C^0(A)^{\N^n}$ belongs to $\cE(A)$ if for all compact subsets $K \subseteq A$, all $p \in \N$, and all $|\al| \le p$

\[
(R^p_xF)^\al(y) = o(|x-y|^{p-|\al|}) \quad \text{ as } |x-y| \to 0, ~ x,y \in K,
\]

\[
(R^p_xF)^\al(y) := F^\al(y) - \sum_{|\be|\le p -|\al|} \frac{(y-x)^\be}{\be!} F^{\al+\be}(x).
\]

Whitney's extension theorem [92] states that $j^\infty_A  C^\infty(\R^n)=\cE(A)$, i.e.,
every Whitney jet $F \in \cE(A)$ admits an extension $f \in C^\infty(\R^n)$ such that $j^\infty_Af = F$.

In this paper we shall be interested in quantitative versions of Whitney's extension theorem.

Given that a Whitney jet $F \in \cE(A)$ satisfies certain uniform growth properties,
can these properties be preserved by the extension?
If they cannot be preserved, can the loss of regularity be controlled?
Can the extension be performed by a continuous linear map?

The uniform growth properties we have in mind are bounds on the multisequence of partial derivatives imposed in terms of a suitable weight
sequence which measures the deviation from the Cauchy estimates and hence from analyticity.
They give rise to so-called ultradifferentiable classes
which form scales of regularity classes between the real analytic and the smooth class.
More specifically, the use of a weight sequence $M$ leads to Denjoy–Carleman classes which originated around 1900
in work of Borel, Gevrey, Holmgren, Hadamard, etc. A different approach based on decay properties of the Fourier transform is due to
Beurling and Björck in the 1960s. An equivalent description of this second approach was later given by Braun, Meise, and Taylor.
The resulting classes are called Braun–Meise–Taylor classes; their definition involves a weight function $\om$.

In this survey we will first treat the extension problem extensively for Denjoy–Carleman classes.
In a second part we will discuss it for Braun–Meise–Taylor classes. Our treatment of the latter is based on
their description as intersections and unions of Denjoy–Carleman classes in terms of an associated
family of weight sequences (called weight matrix).

Our study of <Ref> will first focus on the simplest case when the closed set $A$ is just the singleton $\{0\}$.
Then $j^\infty_{\{0\}}$ is called Borel map, $\cE(\{0\})$ is isomorphic to
the ring of formal power series $\K[[X_1,\ldots,X_n]]$ (where $\K$ is $\R$ or $\C$), and Whitney's extension theorem reduces to Borel's lemma.
Émile Borel's discovery at the end of the 19th century
that there exist classes of smooth functions which have an analytic continuation property (called quasianalyticity),
but contain functions that are nowhere analytic, led to a lot of activity. A class of smooth functions is quasianalytic
if the restriction of the Borel map to this class is injective. The famous Denjoy–Carleman theorem characterizes quasianalyticity
of Denjoy–Carleman classes in terms of the weight sequence $M$. We give a full proof based on two
elementary lemmas (<Ref> and <Ref>) which
will turn out to be foundational for the whole theory.
For example, we will deduce the characterization of quasianalyticity for
Braun–Meise–Taylor classes from the Denjoy–Carleman theorem.

The bounds on the derivatives defining an ultradifferentiable class naturally
induce corresponding bounds for the infinite jet of such functions at $0$.
Thus the natural codomain of the Borel map on an ultradifferentiable class is a sequence space defined by the corresponding bounds,
and one may ask about surjectivity of this map.
We will see that surjectivity on Denjoy–Carleman classes is equivalent to a condition on the weight sequence $M$
that is called strong non-quasianalyticity.
It will turn out that this condition is indispensable for the existence of suitable (even optimal) cutoff functions
crucial for the solution of the extension problem.
By different methods (studying the distribution of zeros of quasianalytic functions and their derivatives)
we shall also see that the Borel map is never surjective in the quasianalytic setting,
except for the real analytic class.

All ultradifferentiable classes considered in this paper come in two types,
Beurling type and Roumieu type.
The classes of Beurling type carry a natural locally convex topology that make them to Fréchet spaces, the
topology of the classes of Roumieu type is more complicated.
We shall see that the Borel map on Denjoy–Carleman classes of Beurling type is even split surjective if
$M$ is strongly non-quasianalytic. In the Roumieu case, generally, the Borel map does not admit a continuous linear

Having solved the extension problem for the Borel map, the solution of the general problem for arbitrary
non-empty closed subsets $A$ of $\R^n$ depends on the existence of optimal cutoff functions.
They are optimal in the sense that they realize necessary sharp bounds.
In conjunction with a family of Whitney cubes for $A$, they yield optimal partitions of unity with the help
of which the local extensions (provided by the solution for the Borel map) can be glued to a global extension
of Whitney ultrajets on $A$. The construction of these cutoff functions involves some delicate properties of
auxiliary functions associated with the weight sequence $M$.

The existence of extensions in the Beurling case follows by a reduction argument from the Roumieu case.
But we will also find optimal cutoff functions of Beurling type and utilize them for a direct proof of the
split surjectivity of the restriction map $j^\infty_A$ on Denjoy–Carleman classes of Beurling type.
In this way we give an elementary constructive proof for the existence of extension operators in this setting,
without relying on the abstract splitting theorem for Fréchet spaces, which we shall however discuss briefly.

If strong non-quasianalyticity is lacking, and hence extension preserving the class is impossible,
one is led to the problem of controlling the loss of regularity.
This problem is solved by describing the pairs
of weight sequences $(M,N)$ (later called admissible pairs)
such that Whitney jets on a closed set $A$ satisfying $M$-bounds admit extensions with $N$-bounds, both in the Beurling and
Roumieu case. Technically, this requires an interesting tool inspired by Dyn'kin's theory of almost analytic functions:
instead of the local extensions (for the singleton) one uses the Taylor polynomials of the jet
to higher and higher degree as the closed set $A$ is approached.

The second part of the survey is dedicated to a concise introduction to Braun–Meise–Taylor classes
and a discussion of the extension problem in that framework.
Our approach is based on the description of these classes as suitable intersections and unions of Denjoy–Carleman classes
which allows (to a fair extent) for a swift and elementary treatment, building on the extensive study of the latter in part one.
The discussion of the extension problem for Braun–Meise–Taylor classes will be more expository (compared to part one).
We will present the state of the art of this area mostly without proofs but indicating
the important ideas and methods involved.
Interestingly, there appear phenomena that are not present in the framework of Denjoy–Carleman classes.
For instance, if extensions preserving the class are possible, even in the Beurling case
they cannot always be realized by extension operators.
The existence of extension operators (always in the Beurling case) depends on the geometry of the set $A$ and
on the weight function $\om$.
The singleton $\{0\}$, for example, admits an extension operator if and only if $\om$ has a certain additional property
(namely, it is a so-called (DN)-weight).
And, if $\om$ has this property, then every closed set $A$ has an extension operator.
Given that $\om$ lacks that property, then the existence of an extension operator on a compact set $K$ is characterized by the
linear topological invariant (DN) of the space of Whitney $\om$-ultrajets of Beurling type on $K$.
A compact set may or may not satisfy this condition: sets with real analytic boundary do
and sets with sharp (i.e., infinitely flat) cusps do not.

In order to make the exposition not too technical we do not strive for the utmost generality of the results.
But we provide appropriate references for the interested reader.

This survey article arose from notes for a mini-course I gave at the
School of Real Geometry in Fortaleza, Brazil, May 24-28, 2021.[<https://sites.google.com/view/scregefor2020/>]

§.§ Notation

We use standard multiindex notation.

The supremum norm is denoted by $\|u\|_K := \sup_{x\in K} |u(x)|$.

A sequence $(a_k)$ is called increasing if $a_k \le a_{k+1}$
and strictly increasing if $a_k<a_{k+1}$ for all $k$;
analogously with decreasing. That an increasing sequence $(a_k)$ tends to infinity
is abbreviated by $a_k \nearrow \infty$.

For two non-negative sequences $a=(a_k)$ and $b=(b_k)$ we write $a \lesssim b$ if there is a constant $C>0$
such that $a_k \le C b_k$ for all $k$; similarly for functions.
We will also use $a \preceq b$ for $a_k^{1/k} \lesssim b_k^{1/k}$ and
$a \lhd b$ for $a_k^{1/k}/ b_k^{1/k} \to 0$.

The indicator function of a subset $A \subseteq \R^n$ is denoted by $\mathbf{1}_A$.
If $A$ is non-empty, then $\on{diam} A := \sup\{|a- b| : a,b \in A\}$ is the diameter of $A$.
If $B \subseteq \R^n$ is another non-empty set, then
$\on{dist}(A,B) := \inf\{|a-b| : a \in A, \, b \in B\}$ is the Euclidean distance between $A$ and $B$.
In particular, $d_A(z) = d(z,A) = \on{dist}(z,A) := \on{dist}(\{z\},A)$ for $z \in \R^n$.

For open subsets $A,B \subseteq \R^n$ we use the symbol $A \Subset B$ to indicate that $A$ is relatively compact in $B$.
And we write $K \subseteq_{cp} A$ if $K$ is a compact subset of $A$.

Denjoy–Carleman classes

§ ULTRADIFFERENTIABLE FUNCTIONS

In this section we discuss natural growth conditions for the infinite sequence of derivatives of smooth functions.

§.§ Denjoy–Carleman classes

Let $M= (M_k)_{k \in \mathbb{N}}$ be a sequence of positive real numbers.
Let $U$ be an open subset of $\mathbb{R}^n$.
For $f \in C^\infty(U)$, $\rho >0$, and compact $K \subseteq U$ we consider
the seminorm

\[
\|f\|^M_{K,\rho} := \sup_{x \in K} \sup_{\alpha \in \mathbb{N}^n} \frac{|f^{(\alpha)}(x)|}{\rho^{|\alpha|} M_{|\alpha|}}
= \sup_{\alpha \in \mathbb{N}^n} \frac{\|f^{(\alpha)}\|_K}{\rho^{|\alpha|} M_{|\alpha|}},
\]

where $\|u\|_K := \sup_{x\in K} |u(x)|$ denotes the supremum norm.
(In the definition of $\|f\|^M_{K,\rho}$ it is not important that $K$ is compact; we will occasionally use $\|f\|^M_{U,\rho}$ for
open sets $U$.)
We define the Denjoy–Carleman class of Beurling type

\begin{align*}
\mathcal{E}^{(M)}(U) &:= \big\{f \in C^\infty(U) : \forall K \subseteq_{cp} U ~\forall \rh > 0 : \|f\|^M_{K,\rho} < \infty\big\}
\end{align*}

and the Denjoy–Carleman class of Roumieu type

\begin{align*}
\mathcal{E}^{\{M\}}(U) &:= \big\{f \in C^\infty(U) : \forall K \subseteq_{cp} U ~\exists \rh > 0 : \|f\|^M_{K,\rho} < \infty\big\}.
\end{align*}

It is convenient to consider also the global Denjoy–Carleman classes

\begin{align*}
\mathcal{B}^{(M)}(U) &:= \big\{f \in C^\infty(U) : \forall \rh > 0 : \|f\|^M_{U,\rho} < \infty\big\},
\\
\mathcal{B}^{\{M\}}(U) &:= \big\{f \in C^\infty(U) : \exists \rh > 0 : \|f\|^M_{U,\rho} < \infty\big\},
\end{align*}

as well as the Banach space $\cB^{M}_\rh(U) = \{f \in C^\infty(U) : \|f\|^M_{U,\rho} < \infty\}$.
We consider the natural locally convex topologies
on these spaces, i.e.,

\begin{align*}
\mathcal{B}^{(M)}(U) &= \on{proj}_{n\in \N} \cB^{M}_{1/n}(U), \quad
\mathcal{B}^{\{M\}}(U) = \on{ind}_{n\in \N} \cB^{M}_n(U),
\end{align*}

\begin{align*}
\cE^{[M]}(U) = \on{proj}_{V \Subset U} \mathcal{B}^{[M]}(V).
\end{align*}

For notational convenience we use $\mathcal{E}^{[M]}$  as placeholder for either $\mathcal{E}^{(M)}$
or $\mathcal{E}^{\{M\}}$
with the understanding that if the placeholder appears repeatedly in a statement then it must be interpreted by
either $\mathcal{E}^{(M)}$ or $\mathcal{E}^{\{M\}}$ at all instances.
In an analogous fashion we use $\cB^{[M]}$, $\cD^{[M]}$, etc.

The spaces $\cB^{(M)}(U)$ and $\cE^{(M)}(U)$ are Fréchet spaces, $\cB^{\{M\}}(U)$ is a Silva space,
and all spaces are nuclear provided that the weight sequence $M$ is derivation-closed, see <Ref>.

Furthermore, we need the subspaces consisting of functions with compact support: for compact $K \subseteq \R^n$ let

\[
\cD^{[M]}(K) := \{f \in \cE^{[M]}(\R^n) : \on{supp} f \subseteq K\}
\]

with the induced topology.
If $U \subseteq\R^n$ is open,
we set

\[
\cD^{[M]}(U) := \on{ind}_{K\subseteq_{cp} U} \cD^{[M]}(K).
\]

We shall see below that $\cD^{[M]}(U)$ can be trivial.

For the sequence $M_k = k!$, the Roumieu class $\mathcal{E}^{\{M\}}(U)$ coincides with the class of real analytic functions $C^\omega(U)$ and
the Beurling class $\mathcal{E}^{(M)}(U)$ consists of the restrictions to $U$ of the entire functions $\cH(\C^n)$ on $\mathbb{C}^n$.
This follows from the Cauchy estimates. In general the sequence $M$ describes a deviation from the Cauchy estimates.

§.§ Weight sequences

It is convenient to require some basic mild regularity conditions for the sequence $M$.
Most importantly we assume that $M$ is
log-convex, that is $(\log M_k)_k$ is a convex sequence.[Generally, this assumption can be made without loss of generality.
Indeed, if $M$ is a positive sequence, then
$\mathcal{E}^{\{M\}}(U) = \mathcal{E}^{\{\underline M\}}(U)$
and $\mathcal{E}^{(M)}(U) = \mathcal{E}^{(\underline M)}(U)$,
where $\ul M$ is the log-convex minorant of $M$,
provided that the classes contain the real analytic class, respectively.
This follows from the Cartan–Gorny inequality, see [41, 26] and <cit.>.
Equivalently, the associated sequence $\mu$ with

\[
\mu_k := \frac{M_k}{M_{k-1}}, \quad k \ge 1,
\]

is increasing. We also assume $\mu_0:= 1 \le \mu_1$.

We call a positive log-convex sequence $M=(M_k)$ with $M_0= 1 \le M_1$ and $M_k^{1/k} \to \infty$
a weight sequence.

Log-convexity of the sequence $m$ defined by

\[
m_k := \frac{M_k}{k!}
\]

is a stronger useful property; it means that $\mu^*$ with $\mu^*_0:=1$ and

\begin{equation*}
\mu_k^* := \frac{m_k}{m_{k-1}} = \frac{\mu_k}{k}, \quad k\ge 1,
\end{equation*}

is increasing.
A weight sequence $M$ with this property is said to be strongly log-convex.

A positive log-convex sequence $M$ with $M_0=1 \le M_1$ has the following properties:

* $M_k^{1/k} \le \mu_k$ for all $k \ge 1$.

* The sequences $(M_k)$ and $(M_k^{1/k})$ are increasing.

* $M_j M_k \le M_{j+k}$ for all $j,k \in \N$.

* $M_k^{1/k} \to \infty$ if and only if $\mu_k \to \infty$.

By assumption, we have $1 \le \mu_1 \le \mu_2 \le \cdots$ and thus $M_k^{1/k} = (\mu_1 \mu_2 \cdots \mu_k)^{1/k} \le \mu_k$, that is (1).
An easy computation shows that $M_k^{1/k} \le M_{k+1}^{1/(k+1)}$ is equivalent to $M_k^{1/k}\le \mu_{k+1}$. So (2) follows.
To see (3) observe that
$M_jM_k\le (\mu_1\cdots\mu_j)(\mu_1\cdots\mu_k) \le \mu_1 \cdots \mu_{j+k} = M_{j+k}$.
Let us check (4).
That $M_k^{1/k} \to \infty$ implies $\mu_k \to \infty$ follows from (1).
If $\mu_k \to \infty$, then for each positive integer $n$ there is $k_n$ such that $\mu_k \ge n$ for all $k \ge k_n$. Then

\[
M_{2k_n} =  M_{k_n} \mu_{k_n+1} \cdots \mu_{2k_n} \ge n^{k_n}
\]

and thus $M_{2k_n}^{1/(2k_n)} \ge  \sqrt{n}$ for all $n$. So $M_k^{1/k} \to \infty$, since $M_k^{1/k}$ is increasing.

§.§ Characteristic functions

Every Roumieu class contains functions whose derivatives
max out the defining bounds.

If $M=(M_k)$ is a weight sequence, then there exists $f \in \cB^{\{M\}}(\mathbb{R},\mathbb{C})$
such that $f^{(k)}(0) = i^k a_k$ with $a_k \ge M_k$ for all $k \in \mathbb{N}$ and $i= \sqrt{-1}$.

The following construction is due to [3], see also <cit.>.
Since $\mu_k$ is increasing, we have

\[
\mu_{k+1}^{j-k} \le \frac{M_j}{M_k}, \quad \text{ for all } (j,k) \in \mathbb{N}^2.
\]

Then the function

\[
f(x) := \sum_{k = 0}^\infty \frac{M_k}{(2 \mu_{k+1})^k} e^{2 i \mu_{k+1} x}
\]

has the required properties. Indeed, $j$-fold term-wise differentiation yields

\[
\Big|\sum_{k = 0}^\infty \frac{M_k}{(2 \mu_{k+1})^k} (2 i \mu_{k+1})^j  e^{2 i \mu_{k+1} x}\Big|
\le M_j \sum_{k = 0}^\infty 2^{j-k}  = 2^{j+1} M_j
\]

whence the series is uniformly convergent and $f\in \cB^{\{M\}}(\mathbb{R}, \mathbb C)$. Moreover,

\[
\frac{f^{(j)}(0)}{i^j} =  \sum_{k = 0}^\infty \frac{M_k}{(2 \mu_{k+1})^{k-j}}  \ge M_j.
\]

The lemma is proved.

§.§ Inclusion relations

Let $M=(M_k)$ and $N= (N_k)$ be positive sequences.
It follows from the definition that the inclusion $\mathcal{E}^{[M]}(U) \subseteq \mathcal{E}^{[N]}(U)$ holds
for all open $U \subseteq \mathbb{R}^n$ and all $n \ge 1$, provided that the positive sequences $M$ und $N$ satisfy

\[
\sup_{k \in \mathbb{N}} \Big(\frac{M_k}{N_k} \Big)^{1/k} < \infty.
\]

In that case we write $M \preceq N$.
Notice that the inclusion $\mathcal{E}^{(M)}(U) \subseteq \mathcal{E}^{\{M\}}(U)$ is trivially true.
It is also clear from the definition that

\[
\lim_{k \to \infty} \Big(\frac{M_k}{N_k} \Big)^{1/k} = 0
\]

implies the inclusion $\mathcal{E}^{\{M\}}(U) \subseteq \mathcal{E}^{(N)}(U)$
for all open $U \subseteq \mathbb{R}^n$, $n \ge 1$.
We abbreviate this relation by $M \lhd N$MN@$M \lhd N$.

Let $M$ be a weight sequence and $N$ a positive sequence. Then:

* The inclusion $\mathcal{E}^{[M]}(U) \subseteq \mathcal{E}^{[N]}(U)$ for all open $U \subseteq \mathbb{R}^n$, $n\ge 1$, is equivalent to $M \preceq N$.

* The inclusion $\mathcal{E}^{\{M\}}(U) \subseteq \mathcal{E}^{(N)}(U)$ for all open $U \subseteq \mathbb{R}^n$, $n\ge 1$, is equivalent to $M \lhd N$.

* The inclusion $\mathcal{E}^{(M)}(U) \subseteq \mathcal{E}^{\{N\}}(U)$ for all open $U \subseteq \mathbb{R}^n$, $n\ge 1$, is equivalent to $M \preceq N$.

Actually, for the necessity of $M \preceq N$, respectively $M \lhd N$,
it is enough to have the respective inclusion relation for some non-empty open subset $U$
of $\mathbb{R}$.

If the smaller space is of Roumieu type, then the necessity of the various conditions follows easily from <Ref>.
That $\mathcal{E}^{(M)}(\R) \subseteq \mathcal{E}^{(N)}(\R)$ and $\mathcal{E}^{(M)}(\R) \subseteq \mathcal{E}^{\{N\}}(\R)$
imply $M \preceq N$, respectively, can be shown by an argument of [21] based on the closed graph theorem.

Let $M$ and $N$ be weight sequences.
The lemma implies that $\mathcal{E}^{[M]} = \mathcal{E}^{[N]}$ if and only if
$M \preceq N \preceq M$. In that case we say that the weight sequences $M$ and $N$ are
(Here (and below) $\mathcal{E}^{[M]} \subseteq \mathcal{E}^{[N]}$ means that the inclusion
$\mathcal{E}^{[M]}(U) \subseteq \mathcal{E}^{[N]}(U)$ holds for all open subsets $U \subseteq \mathbb R^n$, $n \ge 1$.)

In view of $C^\om(U) \cong \cE^{\{(k!)_k\}}(U)$ and $\cH(\C^n) \cong \cE^{((k!)_k)}(U)$ one easily deduces the
following corollary.

Let $M$ be a positive sequence. Then:

* The inclusions $C^\omega(U) \subseteq \mathcal{E}^{\{M\}}(U)$ and
$\mathcal{H}(\mathbb{C}^n) \subseteq \mathcal{E}^{(M)}(U)$ for all open $U \subseteq \mathbb{R}^n$, $n \ge 1$,
are both equivalent to the condition

\[
\liminf_{k\to \infty} m_k^{1/k} > 0.
\]

* The inclusion $C^\omega(U) \subseteq \mathcal{E}^{(M)}(U)$ for all open $U \subseteq \mathbb{R}^n$, $n \ge 1$,
is equivalent to the condition

\[
\lim_{k\to \infty} m_k^{1/k} = \infty.
\]

In this case the inclusion $C^\omega(U) \subseteq \mathcal{E}^{(M)}(U)$ is strict.

§.§ Stability properties

Under suitable assumptions on the weight sequence, Denjoy–Carleman classes are stable under basic operations of smooth analysis.
Let $M=(M_k)$ be a weight sequence.

Pointwise multiplication Let $U \subseteq \R^n$ be open. Then $\cE^{[M]}(U)$ forms a ring with respect to pointwise
multiplication of functions. This follows from the Leibniz rule and <Ref>(3).

Analytic change of variables Suppose that $C^\om(U) \subseteq \cE^{[M]}(U)$ and $\vh : V \to U$ is real analytic, $V \subseteq \R^m$ open.
Then the pullback $\vh^* : \cE^{[M]}(U) \to \cE^{[M]}(V)$, $\vh^*(f) = f\o \vh$, is well-defined; cf. [40].
Hence one can consider $\cE^{[M]}$-functions on real analytic manifolds.

For the following properties the weight sequence $M$ must satisfy additional conditions.

Stability under differentiation $\mathcal{E}^{[M]}(U)$ is stable under differentiation
(i.e. $\partial^\al \mathcal{E}^{[M]}(U) \subseteq \mathcal{E}^{[M]}(U)$ for all $\al \in \mathbb{N}^n$)
if and only if

\begin{equation} \label{eq:der-closed}
\sup_{k\ge 1} \Big(\frac{M_{k+1}}{M_k}\Big)^{1/k} < \infty.
\end{equation}

In this case we say that $M$ is derivation-closed.
It is easy to see that (<ref>) implies that
there is a constant $C\ge 1$ such that $M_k \le C^{k^2}$ for all $k$.
Actually, also the converse implication holds if $M$ is log-convex, see [58].

Stability under composition $\cE^{[M]}$ is stable under composition (i.e.,
$\mathcal{E}^{[M]}(U) \o \mathcal{E}^{[M]}(V,U) \subseteq \mathcal{E}^{[M]}(V)$ for all open sets
$U \subseteq \mathbb{R}^n$ and $V \subseteq \mathbb{R}^m$) provided that

\begin{equation}
\label{eq:FdB}
\preceq M,
\end{equation}

where $m_k^\o := \max\{m_j m_{\al_1} \cdots m_{\al_j} : \al_i \in \mathbb{N}_{>0}, ~ \al_1 + \cdots + \al_j = k\}$. This follows easily from
Faà di Bruno's formula; cf. [74].[It was shown in [74] that (<ref>) is also necessary
for the stability under composition of $\mathcal{E}^{[M]}$ provided that the class is stable by differentiation; this is based on the characterization of inverse-closedness and
stability of $\mathcal{E}^{[M]}$ under superposition by entire functions due to [81, 21].]
We call (<ref>) the (FdB)-property.

The (FdB)-property is not easy to check, but it follows from several conditions that are easier to handle.
If $M$ is a positive sequence, then each of the following conditions implies that $M$ has the (FdB)-property:

* $m=(m_k)$ is log-convex (i.e. $M$ is strongly log-convex).

* $m_j m_k \le m_1 m_{j+k-1}$ for all $j,k \ge 1$.

* $M$ is derivation-closed and the sequence $m_k^{1/k}$ is almost increasingalmost increasing, i.e.,
there exists $C>0$ such that $m_j^{1/j} \le C m_k^{1/k}$ for all $j \le k$.

Conversely, if $M$ is a weight sequence having the (FdB)-property, then
$m_k^{1/k}$ is almost increasing; cf. [74].
See <cit.> for an example of a weight sequence $M$ such that $\cE^{[M]}$ is stable under composition, but there
is no strongly log-convex sequence $N$ that is equivalent to $M$.

Inverse mappings For any $\cE^{[M]}$-mapping $f : U \to V$, where $U \subseteq \R^m$ and $V \in \R^n$ are open,
such that $f'(x_0) \in L(\R^m,\R^n)$ is invertible at $x_0 \in U$ there exist neighborhoods $x_0 \in U_0 \subseteq U$ and
$f(x_0) \in V_0 \subseteq V$ and a $\cE^{[M]}$-mapping $g : V_0 \to U_0$ such that $f \o g = \on{id}_{V_0}$, provided that
$C^\om \subseteq \cE^{[M]}$, $M$ is derivation-closed, and $m_k^{1/k}$ is almost increasing.
Under these assumptions we evidently also have the implicit mapping theorem in $\cE^{[M]}$.

Solving ODEs For any $\cE^{[M]}$-mapping $f : \R \times \R^n \to \R^n$ the solution
of the initial value problem $x' = f(t,x)$, $x(0)=x_0$, is of class $\cE^{[M]}$, where it exists,
provided that
$C^\om \subseteq \cE^{[M]}$, $M$ is derivation-closed, and $m_k^{1/k}$ is almost increasing.

It was proved in [75] that, given that
$C^\om \subseteq \cE^{[M]}$ and $M$ is derivation-closed, the condition that $m_k^{1/k}$ is almost increasing is also necessary for
stability under composition, inverse mappings, and solving ODEs, respectively.
As a consequence $\cE^{[M]}$ is inverse-closed, that is $1/f \in \cE^{[M]}(U)$ if $f \in \cE^{[M]}(U)$ is non-vanishing.

§.§ Moderate growth

Let us finish this section by briefly discussing another property which
often plays a decisive role.
We say that a weight sequence $M$ has moderate growth if

\begin{equation} \label{eq:mg}
\exists C>0 ~\forall j,k \in \N : M_{j+k} \le C^{j+k}M_j M_k.
\end{equation}

Evidently, (<ref>) entails (<ref>). It is not hard to see that (<ref>) is equivalent to $\mu_{k+1} \lesssim M_k^{1/k}$
and in turn to $\mu_{2k} \lesssim \mu_k$; cf. <cit.>. In terms of the spaces $\cE^{[M]}$, moderate growth of $M$ is equivalent to
separativity [59], validity of the exponential law [48, 49, 50], and
stability under ultradifferential operators [47], respectively.
Note that (<ref>) holds for $M$ if and only if it holds for $m$.

The moderate growth condition is rather restrictive. Indeed, it implies that there is a constant $C>0$ such that $\mu_{2^j} \le C \mu_{2^{j-1}}$ for all $j$, and
thus, if $2^j \le k < 2^{j+1}$,

\[
\mu_k \le \mu_{2^{j+1}} \le C \mu_{2^j} \le C^{j+1} \le C^{j+1}\frac{k^s}{2^{js}}
\]

for any $s\ge 1$. If we choose $s\ge 1$ such that the sequence $(C^{j+1}/2^{js})_j$ is bounded, we find that $M \preceq G^s$, where
$G^s_k := k!^s$.

For $s\ge 1$ the sequence $G^s = (G^s_k)$ is a strongly log-convex weight sequences of moderate growth.
It is called the
Gevrey sequence of index $s$.
We have $\cE^{\{G^1\}} = C^\om$. For
$s>1$ a typical function in $\cE^{\{G^s\}}(\R)$ is $t \mapsto \exp(- 1/t^{1/(s-1)})$.
It is easy to check that $(k^{sk})_k$ is an equivalent weight sequence.

§ THE BOREL MAP AND THE DENJOY–CARLEMAN THEOREM

Coming back to <Ref> we shall now discuss our problem for the singleton $\{0\}$.
Suppose that $M$ is a weight sequence and $f \in \cE^{[M]}(U)$, where $U$ is an open connected neighborhood
of $0 \in \R^n$. Then the jet $F:= j^\infty_{\{0\}}f = (f^{(\al)}(0))_{\al \in \N^n}$
obviously satisfies

\begin{equation}
|F|^M_{\rh}  := \sup_{\al \in \N^n} \frac{|F^{\al}|}{\rh^{|\al|} M_{|\al|}} < \infty
\end{equation}

for some $\rh>0$ or for all $\rh>0$, depending on whether we consider the Roumieu case $\cE^{\{M\}}$ or the Beurling case $\cE^{(M)}$.
We define

\begin{align*}
\La^{\{M\}}_n &:= \{a \in \C^{\N^n} : ~\exists \rh>0 : |a|^M_\rh <\infty\},
\\
\La^{(M)}_n &:= \{a \in \C^{\N^n} : ~\forall \rh>0 : |a|^M_\rh <\infty\},
\end{align*}

and equip them with their natural locally convex topology.
In accordance with <Ref>, we use $\La^{[M]}_n$ for both of them and
just write $\La^{[M]}$
if the dimension $n$ is clear from the context.
Now the (restriction of the) Borel map

\begin{equation} \label{eq:Borel}
j^\infty_{\{0\}} : \cE^{[M]}(U) \to \La^{[M]}
\end{equation}

is well-defined.
We may specify <Ref> in this setting and ask:

When is the Borel mapping (<ref>) surjective or injective?

We start with discussing injectivity.

§.§ Quasianalyticity

Let $M$ be a weight sequence.
We say that the class $\cE^{[M]}$ is quasianalytic
if for all open connected neighborhoods $U$ of $0 \in \mathbb{R}^n$ the Borel map $j^\infty_{\{0\}} : \cE^{[M]}(U) \to \La^{[M]}$ is injective.
The class $\cE^{[M]}$ is called non-quasianalytic if it is not quasianalytic.
In that case it is easy to construct non-trivial functions with compact support (so-called bump functions) in the class,
since $\cE^{[M]}$ is stable by multiplication.[The study of quasianalytic classes started
at the end of the nineteenth century with the work of
Émile Borel who found examples of non-trivial sets $\mathcal{C}$ of smooth functions on $\mathbb{R}$ containing nowhere analytic
elements such that all members $f$ of $\mathcal{C}$ have the unique continuation property
$\big(\forall k \in \mathbb{N} : f^{(k)}(0) = 0\big) \implies f =0$.]

There is nothing special about the origin.
It is easy to see that $\cE^{[M]}$ is quasianalytic if and only if
$j^\infty_{\{a\}} : \cE^{[M]}(V) \to \La^{[M]}$ is injective,
where $V \subseteq \R^n$ is any open connected neighborhood of an arbitrary point $a$.

Note that it suffices to check in dimension one whether a class $\cE^{[M]}$ is quasianalytic or not.
Indeed, if there exists	a bump function $f$ in dimension one. Then $f \otimes \cdots \otimes f$ ($n$ times)
is a bump function in dimension $n$.
And, if a function $f$ defined in a neighborhood of $0 \in \R^n$ satisfies
$j^\infty_{\{0\}} f =0$,
then $j^\infty_{\{0\}} (f\o \ell) =0$ for all lines $\ell$ through $0$.

In a communication [42] to the Société Mathématique de France
Jacques Hadamard asked whether
quasianalyticity can be characterized in terms of a growth condition on the iterated derivatives.
This was essentially confirmed a decade later by
[32] and [24, 23]
and became known as the Denjoy–Carleman theorem.
We shall give a proof based on the approach of [31] and <cit.>.

§.§ The size of a function near points of flatness

Let $M=(M_k)$ be a weight sequence.
Let $I = (-r,r) \subseteq \mathbb{R}$ for some $r > 0$.
Let $f \in C^\infty(I)$ be such that $f(t) = 0$ for all $t \le 0$ and
$\|f^{(k)}\|_I \le M_k$ for $0 \le k \le n$.
Then, for all $1 \le \ell \le n$ and $0\le t \le \min\big\{r, \frac{1}{4} \sum_{k = \ell}^n \frac{1}{\mu_k} \big\}$,

\begin{equation} \label{eq:sizenearflat}
|f(t)| \le  \Big( \frac{2 t}{\sum_{k = \ell}^n \frac{1}{\mu_k}}\Big)^\ell.
\end{equation}

Cf. [14] which is based on [3] and [31].
Fix $\ell$ with $1 \le \ell \le n$ and
$t = t_\ell \in (0,r)$.
Let $t_{\ell+1}> t_{\ell+2} > \cdots >  t_{n+1} = 0$ be such that the interval $I_k := [t_{k+1},t_k]$ has length

\[
|I_k| = \frac{a}{\mu_k}, \quad \text{ where } \quad a := \frac{t_\ell}{\sum_{k = \ell}^n \frac{1}{\mu_k}}.
\]

Additionally, we set $I_{n+1} := (-r,0]$.
We define $F(s,k) = \|f^{(s)}\|_{I_k}$ and claim

\begin{align} \label{eq:Fsk}
F(s,k) \le (2 a)^{k-s} M_s \quad \text{ for } \ell \le k \le n+1 \text{ and } 0 \le s \le k.
\end{align}

The claim holds for $s = k$ and for $s \le k = n+1$ by assumption,
and the fundamental theorem of calculus gives the bound

\[
F(s,k) \le F(s,k+1) + |I_k| F(s+1,k).
\]

Let $0 \le s < k \le n$.
If we assume by induction, that (<ref>) is true for $(s,k+1)$ and $(s+1,k)$, then

\[
\le (2 a)^{k+1-s} M_s +  \frac{a}{\mu_k} (2 a)^{k-s-1} M_{s+1}
\le	(2 a)^{k-s} M_s \Big( 2a + \frac12\Big),
\]

since $M_{s+1} = M_s \mu_{s+1} \le M_s \mu_k$.
Provided that $2 a \le 1/2$, claim (<ref>) is proved by induction.
In particular, for $t = t_\ell \in I_\ell$ and $s = 0$, we obtain
\(
|f(t)| \le F(0,\ell) \le (2a)^\ell
\)
which is (<ref>).

As an immediate consequence we observe that the divergence of the series
$\sum_{k} \frac{1}{\mu_k}$
is a
sufficient condition for the injectivity of $j^\infty_{\{0\}} : \cE^{\{M\}}(\R) \to \La^{\{M\}}$.
To see that it is also necessary we aim to construct a non-trivial function with compact support in
$\mathcal{E}^{\{M\}}(\mathbb{R})$ provided that the series converges.

§.§ Special bump functions

For $a>0$ let us consider the step function $H_a := \frac{1}{a} \mathbf 1_{(0,a)}$.
If $f$ is a continuous function on $\mathbb{R}$, then

\[
f * H_a(x) = \frac{1}{a} \int_0^a f(x-t) \, dt = \frac{1}{a} \int_{x-a}^x f(t) \, dt
\]

is $C^1$ with derivative

\[
(f * H_a)'(x) = \frac{f(x) - f(x-a)}{a}.
\]

Hence $f * H_a$ is $C^{k+1}$ provided that $f$ is $C^k$.

Let $(a_k)$ be a decreasing positive sequence with $a := \sum_{k = 0}^\infty a_k < \infty$.
Then $f_k := H_{a_0} * \cdots * H_{a_k}$ is of class $C^{k-1}$ and has support in $[0,a]$.
The sequence $(f_k)$ converges to a $C^\infty$-function $f$ with support in $[0,a]$
such that $\int f \, dx = 1$ and

\[
|f^{(k)}(x)| \le \frac{1}2 \int |f^{(k+1)}(x)| \, dx \le \frac{2^k}{a_0 a_1 \cdots a_k},
\]

for all $k \in \mathbb{N}$ and $x \in \mathbb{R}$.

Cf. <cit.>.
It is easy to see that $f_1$ is the piecewise linear function which vanishes outside $[0,a_0+a_1]$
has slope $(a_0 a_1)^{-1}$ in $[0,a_1]$, is constant in $[a_1,a_0]$ and decreases linearly to $0$ in $[a_0,a_0+a_1]$.
In particular, $f_1$ is continuous.

It follows that $f_k$ is $C^{k-1}$ with support in $[0,a_0 + a_1 + \cdots + a_k]$.
Writing $(\ta_a f)(x) = f(x-a)$ for the translation operator, we have

\[
f_k'(x) = H_{a_0}' * H_{a_1} * \cdots * H_{a_k} (x) = \frac{1}{a_0} (1 - \ta_{a_0}) H_{a_1} * \cdots * H_{a_k} (x)
\]

and iterating we find

\[
f_k^{(j)} = \prod_{i = 0}^{j-1} \frac{1}{a_i} (1 - \ta_{a_i}) H_{a_j} * \cdots * H_{a_k}, \quad \text{ for } j \le k-1.
\]

Since $\int H_{a_i} \, dx = 1$ for all $i$, it follows that

\[
|f_k^{(j)}| \le \frac{2^j}{a_0 a_1 \cdots a_j}  \quad \text{ and } \quad \int |f_k^{(j)}| \, dx
\le \frac{2^j}{a_0 a_1 \cdots a_{j-1}},
\]

where we use $| u * v | \le \sup |u| \int |v| \, dx$ and $\int u * v \,dx = \int u  \,dx \int v \,dx$.
We have

\begin{align*}
|f_{k+\ell}(x)- f_\ell(x)| &= |f_\ell * H_{a_{\ell +1}} * \cdots * H_{a_{\ell +k}}(x) - f_\ell(x)|
\\
&= |\int( f_\ell(x-y) - f_\ell(x)) H_{a_{\ell +1}} * \cdots * H_{a_{\ell +k}}(y) \,dy |
\\
&\le (a_{\ell +1} + \cdots + a_{\ell +k})  \sup |f_\ell'|
\\
&\le 2 \frac{a_{\ell +1} + \cdots + a_{\ell +k}}{a_0 a_1},
\end{align*}

since $H_{a_{\ell +1}} * \cdots * H_{a_{\ell +k}}$ has support in $[0,a_{\ell +1} + \cdots + a_{\ell +k}]$.
It follows that $f_k$ has a uniform limit $f$. Similarly, one sees that all derivatives of $f_k$
have uniform limits.
Hence $f$ is $C^\infty$ and the lemma follows easily.

§.§ The Denjoy–Carleman theorem

We are ready to characterize quasianalyticity in the Roumieu case.
In order to also include the Beurling case we first treat the following lemma.
This lemma will be very useful for reducing the Beurling to the Roumieu case
at several occasions.

Let $(\al_k)_{k\ge 1}$, $(\be_k)_{k\ge 1}$, and $(\ga_k)_{k\ge 1}$ be sequences with the following properties:

* $\al_k \ge 0$ and $\sum_k \al_k<\infty$.

* $\be_k >0$ and $\be_k \to 0$.

* $\ga_k >0$ and $\ga_k \searrow 0$.

Then there exists a positive sequence $(\la_k)_{k\ge 1}$ such that $\la_k \nearrow \infty$ and

* $\sum_{j \ge k} \la_j \al_j \le 8 \la_k \sum_{j \ge k} \al_j$  for all  $k \ge 1$,

* $\la_k \be_k \to 0$,

* $\la_k \ga_k$ is decreasing.

We find a strictly increasing sequence $(j_p)_{p\ge 1}$ of integers with $j_1 =1$ and

\begin{align*}
\sum_{j \ge j_{p+1}} \al_j \le \frac{1}{4} \sum_{j \ge j_{p}} \al_j, \quad
\be_j \le \frac{1}{4^p} ~ \text{ if } j \ge j_{p+1}, \quad
\ga_{j_{p+1}} \le \frac{1}{4} \ga_{j_p}
\end{align*}

for all $p\ge 1$.
Then $2^{p+1} \ga_{j_{p+1}} < 2^p \ga_{j_{p}}< 2^{p+1} \ga_{j_{p}}$. Let $k_p$ be the largest
integer $k$ satisfying $j_p \le k < j_{p+1}$ and $2^p \ga_{j_{p}}\le 2^{p+1} \ga_{k}$.
Define $(\la_k)_{k\ge 1}$ by setting

\[
\la_k :=
\begin{cases}
2^p \frac{\ga_{j_p}}{\ga_{k}} &\text{ if } j_p \le k \le k_{p},
\\
2^{p+1} & \text{ if } k_p < k \le j_{p+1}.
\end{cases}
\]

It is clear that $\la_k \nearrow \infty$ and that $\la_k\ga_k$ is decreasing.
If $j_{p+1} \le k \le j_{p+2}$, then $2^{p+1} \le \la_k \le 2^{p+2}$ and hence
$\la_k \be_k \le  2^{-p+2}$, whence $\la_k \be_k \to 0$.

\begin{align*}
\sum_{j \ge j_{p+1}} \la_j \al_j
=  \sum_{n = 0}^\infty \sum_{j= j_{p+1+n}}^{j_{p+ 2 + n}-1} \la_j \al_j
&\le  \sum_{n = 0}^\infty  2^{p+2 +n}\sum_{j= j_{p+1+n}}^{j_{p+2+n}-1}  \al_j
\\
&\le  \sum_{n = 0}^\infty  2^{p+2+n} \frac{1}{4^{n}} \sum_{j\ge j_{p+1}}  \al_j
= 2^{p+3}  \sum_{j\ge j_{p+1}}  \al_j.
\end{align*}

For $k \ge 1$ take $p$ such that $j_p \le k < j_{p+1}$.

\begin{align*}
\sum_{j \ge k} \la_j \al_j
&= \sum_{j= k}^{j_{p+1}-1} \la_j \al_j + \sum_{j \ge j_{p+1}} \la_j \al_j
\\
&\le 2^{p+1} \sum_{j= k}^{j_{p+1}-1} \al_j + 2^{p+3}  \sum_{j\ge j_{p+1}}  \al_j
\le 8 \cdot 2^{p} \sum_{j\ge  k} \al_j
\le 8 \la_k \sum_{j\ge  k} \al_j.
\end{align*}

The proof is complete.

Let $(\al_k)$ be a decreasing positive sequence with $\sum_k a_k < \infty$.
There exists a decreasing positive sequence $(\be_k)$ such that $\sum_k \be_k < \infty$
and $\be_k/\al_k \nearrow \infty$.

Apply <Ref> to $\al_k = \be_k = \ga_k$. Then $\be_k:= \la_k \al_k$ has the required properties.

Let $M=(M_k)$ be a weight sequence. Then the following conditions are equivalent:

* $\cE^{\{M\}}$ is quasianalytic.

* $\cE^{(M)}$ is quasianalytic.

* $\sum_{k} \frac{1}{\mu_k} = \infty$.

The equivalence of (1) and (3) is an easy consequence of <Ref> and
<Ref> (with $a_k=\mu_k^{-1}$).

If $\cE^{(M)}$ contains non-trivial functions of compact support then so does $\cE^{\{M\}}$,
by the trivial inclusion $\cE^{(M)} \subseteq \cE^{\{M\}}$, which shows (3) $\Rightarrow$ (2)
since we already have the equivalence of (1) and (3).

In order to show that
(2) implies (3) suppose that $\sum_{k} \frac{1}{\mu_k} < \infty$.
By <Ref>, we find an increasing sequence $(\nu_k)$ such that $\sum_k \frac{1}{\nu_k}  < \infty$
and $\frac{\mu_k}{\nu_k} \nearrow \infty$. So the sequence $\frac{M_k}{N_k}$ is log-convex and thus
$(\frac{M_k}{N_k})^{1/k} \to \infty$ since $\frac{\mu_k}{\nu_k} \to \infty$ (cf. <Ref>(4)).
Then $N=(N_k)$ is a weight sequence satisfying $N \lhd M$ and thus $\cE^{\{N\}} \subseteq \cE^{(M)}$.
Since $\sum_k \frac{1}{\nu_k}  < \infty$, we find that $\cE^{\{N\}}$ is non-quasianalytic, and hence so is $\cE^{(M)}$.

The Denjoy–Carleman theorem completely characterizes injectivity of the Borel map (<ref>).
It shows that the following definition makes sense.

A weight sequence $M=(M_k)$ is called quasianalytic if
the equivalent conditions of <Ref> hold, and it is called
non-quasianalytic otherwise.

For later reference note that the proof of the implication (2) $\Rightarrow$ (3) in <Ref> shows the following:
For any non-quasianalytic weight sequence $M$ there exists a non-quasianalytic weight sequence $N$ with $N \lhd M$.

For a weight sequence $M=(M_k)$, the divergence of the series $\sum_k \frac{1}{\mu_k}$ is
equivalent to the divergence of $\sum_k \frac{1}{M_k^{1/k}}$.
This follows from $M_k^{1/k} \le \mu_k$ for all $k$ (cf. <Ref>(1))
and Carleman's inequality:
If $(a_k)$ is a positive sequence, then

\[
\sum_{n=1}^\infty (a_1 a_2 \cdots a_n)^{1/n} \le e \sum_{k=1}^\infty a_k.
\]

Cf. <cit.>.

If $M$ is not necessarily a weight sequence, then one can work with the log-convex minorant

\[
\ul M_k := \inf\{M_k, M_j^{\frac{\ell-k}{\ell-j}} M_\ell^{\frac{k-j}{\ell-j}} : j <k < \ell\}
\]
of $M$
or the increasing minorant

\[
u_k :=\inf_{k \ge j} M_k^{1/k}
\]

of $M_k^{1/k}$. In fact, the following are equivalent (see <cit.>):

* $\cE^{[M]}$ is quasianalytic.

* $\sum_k \frac{1}{u_k} = \infty$.

* $\sum_k \frac{1}{\ul M_k^{1/k}} = \infty$.

* $\sum_k \frac{1}{\ul \mu_k} = \infty$.

(1) The Gevrey sequences $G^s$, $s\ge 1$, are non-quasianalytic if and only if $s>1$.

(2) The Denjoy sequences

\[
Q^{n,\de}_k = k^k \prod_{j=1}^{n-1} \big(\log^{[j]}(k + \exp^{[j]}(1))\big)^{k}
\big(\log^{[n]}(k + \exp^{[n]}(1))\big)^{\de k},
\]

where $n \in \N_{\ge 1}$, $0< \de \le 1$,
and $f^{[n]}$ is the $n$-fold composite of $f$, are quasianalytic strongly log-convex weight sequences
of moderate growth.

§.§ Non-quasianalytic cutoff functions

Before we study surjectivity of the Borel map (<ref>) in the next section
we show that we may use <Ref> to construct non-quasianalytic
cutoff functions in all dimensions.

Let $U \subseteq \mathbb{R}^n$ be open and $K \subseteq U$ compact.
Let $\| \cdot \|$ be any norm on $\mathbb{R}^n$.
If $d = \inf\{ \| x-y\| : x \in K,~ y \in \mathbb{R}^n\setminus U\}$
and $(d_j)$ is a positive decreasing sequence with
$\sum_{j=1}^\infty d_j < d$, then
there exists $\ph \in C^\infty_c(U)$ with $0 \le \ph \le 1$ such that $\ph =1$ on a neighborhood
of $K$ and

\[
\le C(n)^k \frac{\|v_1\| \|v_2\| \cdots \|v_k\|}{d_1 d_2 \cdots d_k}, \quad k \ge 1, ~ x \in U.
\]

Let us first consider the norm $\|x\| = \max_i |x_i|$.
Let $f$ be the function from <Ref> (with $a_j = d_{j+1}$)
and set $h(t) := f(t + \frac{1}{2} \sum_{j=1}^\infty d_j)$.
Then $t \in \on{supp} h$ implies $|t| \le \frac{1}{2} \sum_{j=1}^\infty d_j$,
we have $\int h \, dt = 1$, and

\[
\int |h^{(j)}(t)| \,dt \le \frac{2^j}{d_1 d_2 \cdots d_j}.
\]

Let $\ch(x) := h(x_1) h(x_2) \cdots h(x_n)$ which satisfies $\int \ch \, dx =1$
and set $\ph := \ch * \mathbf 1_{K_{d/2}}$,
where $K_{d/2} := \{y\in \R^n : \|x-y\| \le d/2 \text{ for some } x \in K\}$.
Then $\ph  \in C^\infty_c(U)$ is as required, since

\[
|\ph^{(\al)}| \le  \int |\ch^{(\al)}(x)| \,dx \le \frac{2^{|\al|}}{d_1 d_2 \cdots d_{|\al|}}
\]

as $(d_k)$ is decreasing.
The statement for arbitrary norms follows from the fact that any two norms on $\mathbb R^n$ are
equivalent with constants only depending on $n$.

Let $M$ be a non-quasianalytic weight sequence.
Let $U \subseteq \mathbb{R}^n$ be open and $K \subseteq U$ compact.
Then there exists a function $\ph \in \mathcal{E}^{[M]}(U)$ with support contained in $U$ such that
$\ph = 1$ in a neighborhood of $K$.

Let $d := \on{dist}(K,\mathbb R^n \setminus U)$[The distance
$\on{dist}(A,B) := \inf\{|a-b| : a \in A, b \in B\}$ of two subsets of $\R^n$ is defined with respect
to the Euclidean norm.]
and $d_j := \frac{d}{s\mu_j}$, where $s := \sum_{j\ge 1} \frac{1}{\mu_j} < \infty$.
Then the proposition provides a function $\ph \in \mathcal{E}^{\{M\}}(U)$ with the desired properties.
Using <Ref>,
we may achieve that $\ph \in \mathcal{E}^{(M)}(U)$.

That non-quasianalytic functions are “abundant” is witnessed by the following remark.

Let $M$ be a non-quasianalytic weight sequence.
For each closed subset $A \subseteq \R^n$ there exists $f \in \mathcal{B}^{[M]}(\mathbb{R}^n)$
such that $f$ is flat on $A$, i.e., $j^\infty_A f =0$, and
$f|_{\R^n \setminus A}$ is positive and real analytic. Cf. [55, 56].

§ SURJECTIVITY OF THE BOREL MAP

Let $M=(M_k)$ be a weight sequence. We shall see that a necessary and sufficient condition for
the surjectivity of the Borel map $j^\infty_{\{0\}} : \cE^{[M]}(\R) \to \La^{[M]}$ is

\begin{equation} \label{eq:snq}
\sup_{k} \frac{\mu_k}{k} \sum_{j \ge k} \frac{1}{\mu_j} < \infty.
\end{equation}

In the Beurling case the condition even guarantees that the Borel map is split surjective.
This section is primarily based on [68].

A weight sequence $M$ satisfying (<ref>) is said to be strongly non-quasianalytic.[Many authors use Komatsu's (M.$\ul{\hspace{2mm}}$) notation [47]:
log-convexity (M.1),
moderate growth (M.2),
strong non-quasianalyticity (M.3),
derivation closedness (M.2)',
non-quasianalyticity (M.3)'.]

Obviously a strongly non-quasianalytic weight sequence $M$ is non-quasianalytic, and thus
$\cE^{[M]}$ admits cutoff functions. We will see that (<ref>) allows for
special cutoff functions that are in a certain sense optimal; see <Ref>.

The Gevrey sequences $G^s$ are strongly non-quasianalytic provided that $s>1$.
Indeed, $\ga^s_k = G^s_k/G^s_{k-1} = k^s$ and

\[
\sum_{j \ge k} \frac{1}{j^s} \le \int_{k-1}^\infty \frac{1}{t^s}\, dt
= \frac{1}{s-1} \frac{1}{(k-1)^{s-1}} = \frac{1}{s-1} \Big(\frac{k}{k-1}\Big)^{s-1} \frac{k}{k^s}
\le  \frac{2^{s-1}}{s-1} \frac{k}{k^s}.
\]

§.§ Necessity

Let us assume throughout this section that $M$ is non-quasianalytic.
The quasianalytic case which requires different methods will be discussed in <Ref>.

Let $M$ be a non-quasianalytic weight sequence.
If the Borel map $j^\infty_{\{0\}} : \cE^{(M)}(\R) \to \La^{(M)}$ or
$j^\infty_{\{0\}} : \cE^{\{M\}}(\R) \to \La^{\{M\}}$ is surjective, then $M$ is strongly non-quasianalytic.

We give details in the Beurling case and indicate the required changes for the Roumieu case.

We may suppose that $j^\infty_{\{0\}} : \cB^{(M)}((-1,1)) \to \La^{(M)}$ is surjective,
by multiplication with a suitable cutoff function.
Then the open mapping theorem for Fréchet spaces
(cf. <cit.>) implies that
there exist constants $C,\rh>0$ and functions $f_k \in \cB^{(M)}((-1,1))$ such that $f_k^{(j)}(0) = \de_{jk}$ and

\begin{equation} \label{eq:nec0}
\|f_k\|^M_{[-1,1],1} \le \frac{C}{\rh^k M_k}, \quad k \in \N.
\end{equation}

Let $t_k := \inf\{t \in [0,1] : f_k^{(k)}(t) < \frac{1}{2}\}$. Then $f_k^{(k)}(t)\ge \frac{1}{2}$ for $t \in [0,t_k]$ and hence
\(
f_{2k}^{(k)}(t) \ge \frac{t^k}{2 k!}
\)
for $t \in [0,t_{2k}]$.
In particular, for $t=t_{2k}$ we find, by (<ref>),
\(
\frac{t_{2k}^k}{2 k!} \le f_{2k}^{(k)}(t_{2k}) \le \frac{C M_k}{\rh^{2k} M_{2k}}
\)
which implies (using that $\mu_k$ is increasing)

\begin{equation} \label{eq:nec1}
t_{2k} \le \frac{(2C)^{1/k}}{\rh^2} \frac{k}{\mu_k}.
\end{equation}

We claim that there exists $0<h< \min\{\tfrac{1}{4}, \tfrac{\rh}{4}\}$ such that

\begin{equation} \label{eq:nec2}
t_{k} \ge h \sum_{j\ge 2k} \frac{1}{\mu_j} \quad \text{ for all sufficiently large } k.
\end{equation}

Then (<ref>) and (<ref>) imply (<ref>),

\begin{align*}
\sum_{j\ge k} \frac{1}{\mu_j} = \sum_{j= k}^{4k-1} \frac{1}{\mu_j} + \sum_{j\ge 4k} \frac{1}{\mu_j}
\le \frac{3k}{\mu_k} + \frac{t_{2k}}{h}	\lesssim \frac{k}{\mu_k}.
\end{align*}

It remains to show (<ref>). To this end we apply <Ref> to the functions

\[
g_k(t) :=
\begin{cases}
\big(\frac{C}{\rh^k} + 1\big)^{-1} (f_k^{(k)}(t) -1) & \text{ if } t\ge 0,
\\
0 & \text{ if } t<0.
\end{cases}
\]

Indeed, each $g_k$ is $C^\infty$, satisfies
$|g_k| \le 1$, and for $j\ge 1$ we have

\[
|g_k^{(j)}| = \Big(\frac{C}{\rh^k} + 1\Big)^{-1}  |f_k^{(k+j)}| \le \Big(\frac{C}{\rh^k} + 1\Big)^{-1} \frac{C}{\rh^k}  \frac{M_{k+j}}{M_k}\le \mu_{k+1}\cdots \mu_{k+j}=:M^{+k}_j,
\]

by (<ref>).
Then $M^{+k}$ (with $M^{+k}_{0}:=1$) is a weight sequence and <Ref> gives

\[
|g_k(t)| \le \Big(\frac{2t}{\sum_{j\ge 2k} \frac{1}{\mu_j}}\Big)^k, \quad 0\le t \le \frac{1}{4}\sum_{j\ge 2k} \frac{1}{\mu_j}.
\]

Suppose for contradiction that (<ref>) is violated for infinitely many $k$.
For those $k$ we obtain

\[
\frac{1}{2} \frac{\rh^k}{C+\rh^k} = |g_k(t_k)| \le \Big(\frac{2t_k}{\sum_{j\ge 2k} \frac{1}{\mu_j}}\Big)^k \le (2h)^k,
\]

which is however only possible for finitely many $k$. Thus (<ref>) is proved.

Now suppose that $j^\infty_{\{0\}} : \cB^{\{M\}}((-1,1)) \to \La^{\{M\}}$ is surjective.
By the Grothendieck factorization theorem
(cf. <cit.>),
there exist $C,\rh>0$ with

\[
\{(a_k) : |(a_k)|^M_1 \le 1\} \subseteq j^\infty_{\{0\}} \{ f : \|f\|^M_{[-1,1],\rh} \le C\}.
\]

Thus there exist $f_k \in \cB^{\{M\}}((-1,1))$ such that $f_k^{(j)}(0) = \de_{jk}$ and

\[
\|f_k\|^M_{[-1,1],\rh} \le \frac{C}{M_k}, \quad k \in \N.
\]

It is easy to modify the above proof to conclude again (<ref>).

§.§ Sufficiency

Let $M$ be a strongly non-quasianalytic weight sequence. Then:

* $j^\infty_{\{0\}} : \cE^{\{M\}}(\R) \to \La^{\{M\}}$ is surjective.

* $j^\infty_{\{0\}} : \cE^{(M)}(\R) \to \La^{(M)}$ is split surjective, i.e., there exists a continuous linear right-inverse
$E : \La^{(M)} \to \cE^{(M)}(\R)$ with $j^\infty_{\{0\}} \o E = \on{id}_{\La^{(M)}}$.

By [68] (see also <cit.>),
the existence of a continuous linear right-inverse for
$j^\infty_{\{0\}} : \mathcal{E}^{\{M\}}(\R) \to \Lambda^{\{M\}}$ is equivalent to (<ref>) and additionally

\begin{equation} \label{eq:PetzscheRI}
\forall \ep>0 ~\exists a \in \mathbb{N}_{>1} :
\limsup_{k \to \infty} \Big(\frac{M_{ak}}{M_k}\Big)^{\frac{1}{k(a-1)}} \frac{1}{\mu_{ak}} \le \ep.
\end{equation}

This condition is not easy to satisfy; for instance Gevrey sequences fail it.

\[
\Big(\frac{G^s_{ak}}{G^s_k}\Big)^{\frac{1}{k(a-1)}} \frac{1}{\ga^s_{ak}}
\ge \big((k(a-1))!^s\big)^{\frac{1}{k(a-1)}} \frac{1}{(ak)^s}
\ge \Big(\frac{a-1}{ea}\Big)^s.
\]

See also <Ref>.

Let us concentrate on the more interesting Beurling case (2) in <Ref>.
We have to develop some tools before we can give the proof in <Ref>.

§.§ The descendant of a non-quasianalytic weight sequence

A strongly non-quasianalytic weight sequence is equivalent to a strongly log-convex sequence, as will be shown in the next lemma.
More generally, we may associate with any non-quasianalytic
weight sequence $N$ a weight sequence $S$ with many good properties.
The following construction was used in [78]; it is inspired by <cit.>.

Let $\nu=(\nu_k)$ be an increasing positive sequence with $\nu_0 =1$ and  $\sum_k \frac{1}{\nu_k} < \infty$.
The descendant of $\nu$ is the positive sequence $\si = \si(\nu)$ defined by setting $\si_0 := 1$,

\begin{equation} \label{tau}
\ta_k := \frac{k}{\nu_k} + \sum_{j\ge k} \frac 1 {\nu_j}, \quad k \ge 1,
\end{equation}

\begin{equation} \label{sigma}
\si_k := \frac{\ta_1 k}{\ta_k}, \quad k \ge 1.
\end{equation}

We shall also say that $S_k=\si_0 \si_1 \cdots \si_k$ is the descendant of $N_k = \nu_0 \nu_1 \cdots \nu_k$.

Let $\si$ be the descendant of $\nu$. Then:

* $\sup_k \frac{\si_k}{\nu_k} < \infty$.

* $\sup_{k} \frac{\si_k}k \sum_{j\ge k}  \frac{1}{\nu_j} < \infty$.

* $1 \le  \frac{\si_k}k =: \si^* \nearrow \infty$.

* $\si$ is maximal among the sequences satisfying (1) and (2):
If $\mu$ is an increasing positive sequence with $\sup_k \frac{\mu_k}{\nu_k} < \infty$
and $\sup_k \frac{\mu_k}k \sum_{j\ge k}  \frac{1}{\nu_j} < \infty$, then $\sup_k \frac{\mu_k}{\si_k} < \infty$.

(1), (2), and (3) are immediate.
The assumptions on $\mu$ in (4) imply
\(
\ta_k \lesssim \frac{k}{\nu_k} +  \frac{k}{\mu_k} \lesssim \frac{k}{\mu_k}
\)
and hence $\mu_k \lesssim \si_k$.

Note that, by (3), the descendant $S$ of $N$ is a strongly log-convex weight sequence.

The descendant $\si$ of $\nu$ satisfies

\begin{equation} \label{eq:qequivalent}
\exists C>0 ~\forall k : C^{-1} \le \frac{\si_k}{\nu_k} \le C
\end{equation}

if and only if $N$ is strongly non-quasianalytic.
In that case the weight sequences $S$ and $N$ are equivalent.

This follows from (1), (2), and (4) in <Ref>.

Let $N=(N_k)$ be a strongly non-quasianalytic strongly log-convex weight sequence and $S$ its descendant.
for all small $\ep >0$, $\tilde S := (S_k/k!^\ep)$ is a strongly non-quasianalytic weight sequence.

Note that $\nu_k^* := \frac{\nu_k}k$ is increasing. So, for $\si_k^*  := \frac{\si_k}{k}$ and $k\ge 1$,

\begin{align*}
\frac{\si_{2k}^*}{\si_{k}^*} = \frac{\ta_k}{\ta_{2k}}
=\frac{\frac{1}{\nu_k^*} + \sum_{j\ge k} \frac 1 {\nu_j}}{\frac{1}{\nu_{2k}^*} + \sum_{j\ge 2k} \frac 1 {\nu_j}}
\ge
\frac{\frac{1}{\nu_{2k}^*} + \sum_{j\ge k} \frac 1 {\nu_j}}{\frac{1}{\nu_{2k}^*} + \sum_{j\ge 2k} \frac 1 {\nu_j}}
= 1+  \frac{\sum_{k \le j<  2k} \frac 1 {\nu_j}}{\frac{1}{\nu_{2k}^*} + \sum_{j\ge 2k} \frac 1 {\nu_j}}.
\end{align*}

Since $\sum_{k \le j<  2k} \frac 1 {\nu_j} \ge \sum_{k \le j<  2k} \frac 1 {\nu_{2k}} = \frac k {\nu_{2k}}$
and $\frac{1}{\nu_{2k}^*} + \sum_{j\ge 2k} \frac 1 {\nu_j} \lesssim \frac{1}{\nu_{2k}^*}$, we find that

\[
q:=\inf_{k\ge 1} \frac{\si_{2k}^*}{\si_{k}^*} > 1.
\]

Suppose that $\ep\in (0,1)$ is such that $q>2^{\ep}$.
Then $\tilde \si_k^* := \frac{\tilde \si_k}{k} =  \frac{\si_k^*}{k^{\ep}}$ satisfies

\[
\tilde q:= \inf_{k\ge 1} \frac{\tilde \si_{2k}^*}{\tilde \si_{k}^*} = \inf_{k\ge 1} \frac{\si_{2k}^*}{\si_{k}^*} \cdot \frac{1}{2^\ep}> 1.
\]

The fact that $\si^*_k = \frac{\si_k}{k}$ is increasing implies that $\tilde \si_k$ is increasing. Thus,

\begin{align*}
\sum_{j \ge k} \frac{1}{\tilde \si_j}
= \sum_{\ell =0}^\infty \sum_{j=2^\ell k}^{2^{\ell+1} k-1} \frac{1}{\tilde \si_j}
\le  \sum_{\ell =0}^\infty  \frac{2^\ell k}{\tilde \si_{2^\ell k}}
= \sum_{\ell =0}^\infty  \frac{1}{\tilde \si_{2^\ell k}^*}
\le  \frac{1}{\tilde \si_k^*} \sum_{\ell =0}^\infty \tilde q^{-\ell}.
\end{align*}

It follows that $\tilde S$ is a strongly non-quasianalytic weight sequence.

Let $M=(M_k)$ be a strongly non-quasianalytic weight sequence.
We may assume that $\tilde \mu := (\mu_k/k^\ep)$ is increasing to infinity and satisfies
(<ref>) if $\ep>0$ is small enough. Hence
the sequence

\begin{equation} \label{eq:modifiedseq}
\underbrace{\mu_k,\ldots,\mu_k}_{k \text{ times}}, \mu_{k+1} \big(\tfrac{k}{k+1}\big)^\ep,\mu_{k+2} \big(\tfrac{k}{k+2}\big)^\ep, \ldots
\end{equation}

is increasing and for the sum of the reciprocals we have

\begin{equation} \label{eq:cutoffk}
\frac{k}{\mu_k} + \sum_{j \ge k+1} \frac{1}{\mu_j} \Big(\frac{j}{k}\Big)^\ep
\le A \frac{k}{\mu_k}, \quad  k \ge 1,
\end{equation}

for some constant $A>0$.

§.§ Proof of <Ref>

We give a detailed proof of (2).

For each $k \in \N$, there exists $\ps_k \in \cD^{(M)}(\R)$ such that $\ps_k^{(j)}(0) = \de_{jk}$

\begin{equation} \label{eq:claimsur}
\|\ps_k\|^M_{\R,\rh} \le \frac{C_\rh H_\rh^k}{M_k} \quad \text{ for all small } \rh>0.
\end{equation}

Then $E : \La^{(M)} \to \cE^{(M)}(\R)$ defined by
\(
E: a = (a_k) \mapsto \sum_{k\ge 0} a_k \ps_k
\)
is the required right inverse of $j^\infty_{\{0\}}$.
for every $\si>0$ we have $|a|^M_\si < \infty$. Let $\rh>0$ (small) be given. Then

\begin{align*}
\Big| \sum_{k\ge 0} a_k \ps_k^{(j)} \Big| &\le \sum_{k\ge 0} |a_k| |\ps_k^{(j)}|
\\
&\le  \sum_{k\ge 0} |a|^M_\si\, \si^k M_k \cdot  \|\ps_k\|^M_{\R,\rh}\, \rh^j M_j
\\
&\le  \sum_{k\ge 0} |a|^M_\si \si^k M_k \cdot \frac{C_\rh H_\rh^k}{M_k} \rh^j M_j
= |a|^M_\si  \cdot C_\rh \rh^j M_j \cdot \sum_{k\ge 0}  (\si H_\rh)^k.
\end{align*}

We may choose $\si=\si(\rh)>0$
such that \(\sum_{k\ge 0}  (\si H_\rh)^k\) converges.
It follows that $f := E(a)=\sum_{k\ge 0} a_k \ps_k$ defines a $C^\infty$-function $f$ on $\R$ such that
for all $\rh>0$ there exist $\si>0$ and $C>0$ such that

\begin{equation} \label{eq:extop}
\|f\|^M_{\R,\rh} \le C  |a|^M_\si,
\end{equation}

in particular, $f \in \cE^{(M)}(\R)$. So the linear map $E$ is continuous and $\Bmap E(a) = a$ is clear from $\ps_k^{(j)}(0) = \de_{jk}$.

Let us prove the claim.
We apply <Ref> to the setup in <Ref> (for $K=\{0\}$ and $\sum d_j$ the left-hand side of (<ref>)).
So for each $k \ge 1$ there is a $C^\infty$-function $\vh_k$ with $0 \le \vh_k \le 1$, $\on{supp} \vh_k \subseteq (-A \frac{k}{\mu_k},A \frac{k}{\mu_k})$,
$\vh_k(0)=1$, $\vh_k^{(j)}(0) = 0$ for all $j \ge 1$, and

\begin{align*}
|\vh_k^{(j)}| \le
\begin{cases}
2^j \mu_k^j & \text{ if } 0 \le j \le k,
\\
2^j \mu_k^k \frac{M_j}{M_k} \Big( \frac{k^{j-k} k!}{j!}\Big)^\ep & \text{ if } j>k.
\end{cases}
\end{align*}

Then $\ps_k(t):= \vh_k(t) \frac{t^k}{k!}$ clearly satisfies $\ps_k^{(j)}(0) = \de_{jk}$.
It remains to show (<ref>).
If $0 \le j \le k$, then

\begin{align} \label{eq:est1}
|\ps_k^{(j)}| &\le \sum_{i =0}^j \binom{j}{i} 2^i \mu_k^i \Big(A \frac{k}{\mu_k}\Big)^{k-j+i} \frac{1}{(k-j+i)!}
\notag \\
&\le \frac{M_j}{M_k}  \frac{\mu_{j+1} \cdots \mu_k}{\mu_k^{k-j}} (Ae)^k \sum_{i =0}^j \binom{j}{i} 2^i  A^{-j+i}
\le  \frac{M_j}{M_k} (Ae)^k (2 + A^{-1})^j.
\end{align}

If $j\ge 2 k$, then

\begin{align} \label{eq:est2}
|\ps_k^{(j)}| &\le \sum_{i =j-k}^j \binom{j}{i} 2^i \mu_k^k \frac{M_i}{M_k} \Big(\frac{k^{i-k} k!}{i!}\Big)^\ep \Big(A \frac{k}{\mu_k}\Big)^{k-j+i} \frac{1}{(k-j+i)!}
\notag \\
&\le \frac{M_j}{M_k}  \frac{\mu_k^{j-i}\mu_{k+1} \cdots \mu_i}{\mu_{k+1} \cdots \mu_j} (Ae)^k \Big(\frac{k^{j-2k} k!}{(j-k)!}\Big)^\ep \sum_{i =j-k}^j \binom{j}{i} 2^i A^{-j+i}
\notag \\
&\le \frac{M_j}{M_k} (Ae)^k  (2 + A^{-1})^j \Big(\frac{k^{j-2k} }{(j-2k)!}\Big)^\ep.
\end{align}

If $k < j < 2k$, then (since $\frac{k^{i-k} k!}{i!} = \frac{k^i}{i!} \frac{k!}{k^k} \le e^k$)

\begin{align} \label{eq:est3}
|\ps_k^{(j)}| &\le \sum_{i =0}^k \binom{j}{i} 2^i \mu_k^i \Big(A \frac{k}{\mu_k}\Big)^{k-j+i} \frac{1}{(k-j+i)!}
\notag \\
&\quad +
\sum_{i =k+1}^j \binom{j}{i} 2^i \mu_k^k \frac{M_i}{M_k} \Big(\frac{k^{i-k} k!}{i!}\Big)^\ep \Big(A \frac{k}{\mu_k}\Big)^{k-j+i} \frac{1}{(k-j+i)!}
\notag \\
&\le \frac{M_j}{M_k} (Ae)^k  (2 + A^{-1})^j +
\frac{M_j}{M_k} (Ae^2)^k  (2 + A^{-1})^j
\notag \\
&\le \frac{M_j}{M_k} 2(Ae^2)^k  (2 + A^{-1})^j.
\end{align}

Let $0<\rh < 2 + A^{-1}$ be fixed. For $j < 2k$, we get from (<ref>) and (<ref>) that

\[
\frac{|\ps_k^{(j)}|}{\rh^j M_j} \le \frac{1}{M_k} 2(Ae^2)^k  \Big(\frac{2 + A^{-1}}{\rh}\Big)^j
\le \frac{1}{M_k} 2(Ae^2)^k  \Big(\frac{2 + A^{-1}}{\rh}\Big)^{2k}
\]

and, for $j \ge 2k$,  (<ref>) gives

\begin{align*}
\frac{|\ps_k^{(j)}|}{\rh^j M_j}
&\le \frac{1}{M_k} (Ae)^k  \Big(\frac{2 + A^{-1}}{\rh}\Big)^j \Big(\frac{k^{j-2k} }{(j-2k)!}\Big)^\ep
\\
&= \frac{1}{M_k} (Ae)^k \Big(\frac{2 + A^{-1}}{\rh}\Big)^{2k}  \Big(\frac{2 + A^{-1}}{\rh}\Big)^{j-2k} \Big(\frac{k^{j-2k} }{(j-2k)!}\Big)^\ep
\\
&= \frac{1}{M_k} (Ae)^k \Big(\frac{2 + A^{-1}}{\rh}\Big)^{2k}  \exp\Big(k \ep \Big(\frac{2 + A^{-1}}{\rh}\Big)^{1/\ep} \Big).
\end{align*}

This completes the proof of the claim and hence of (2) in <Ref>.

Let us briefly discuss the Roumieu case, that is (1) in <Ref>.
Given $a=(a_k) \in \La^{\{M\}}$, which means that $|a|^M_\rh<\infty$ for some $\rh>0$, one
looks for functions $\ps_k$, $k \in \N$, such that $f:= \sum_{k \ge 0} a_k \ps$ is of class $\cE^{\{M\}}$ and $\Bmap f = a$.
It suffices to repeat the above construction, where (<ref>) is replaced by

\begin{equation*}
\underbrace{H\mu_k,\ldots,H\mu_k}_{k \text{ times}}, H\mu_{k+1},H\mu_{k+2}, \ldots
\end{equation*}

If $H>0$ is chosen large enough, one finds that there is a constant $B\ge 1$ such that

\[
\|f\|^M_{\R,B H} \le C(\rh) \, |a|^M_\rh,
\]

but $H$ and thus also $\ps_k$ and $f$ depend on $\rh$; cf. <Ref>.

§ THE BOREL MAP IN THE QUASIANALYTIC SETTING

Let us investigate surjectivity of the Borel map in the quasianalytic case.
Since there are no cutoff functions, we will restrict $\Bmap$
to the ring of germs of $\cE^{[M]}$-functions at $0$ which we denote by $\cE^{[M]}_0$.
(The dimension of the ambient space will be clear from the context.)
A result of Carleman [22]
states that the Borel map $\Bmap : \cE^{[M]}_0 \to \La^{[M]}$ is never surjective if $M$ is a quasianalytic weight sequence
with $m_k^{1/k} \to \infty$. Recall that $m_k^{1/k} \to \infty$ means that the real analytic class
is strictly contained in $\cE^{[M]}$; on the ring of real analytic germs the Borel map is
an isomorphism $\cE^{\{(k!)\}}_0 \cong \La^{\{(k!)\}}$.

Many different proofs of Carleman's result are known.
We present a much stronger result
which shows that certain elements of $\La^{[M]}$
cannot be extended to germs of class $\cE^{\{N\}}$, where $N$ is any quasianalytic weight sequence.
It is based on a theorem of Bang [4] on the zeros of quasianalytic functions and their derivatives.
See also [65] and [77].

§.§ Zeros of quasianalytic functions and their derivatives

Let $M$ be a quasianalytic weight sequence.
Suppose that $f \in C^\infty([0,1])$ is not identically zero and satisfies

\begin{equation} \label{eq:Bd1}
\|f^{(k)}\|_{[0,1]} \le M_k \quad  \text{ for all } k \in \N.
\end{equation}

Suppose that for $0\le j \le m$ there exists $x_j \in [0,1]$ such that $f^{(j)}(x_j) = 0$.

\begin{equation} \label{eq:assertion}
\sum_{j=1}^m |x_{j-1} -x_j| \ge \frac{1}{e} \sum_{\ka < k \le m} \frac{1}{\mu_k},
\end{equation}

\begin{equation} \label{eq:kappa}
\ka = - \log \sup_{j \ge 0} \frac{|f^{(j)}(x_0)|}{e^j M_j}.
\end{equation}

For $n \in \N$ and $t \in [0,1]$ consider
\(
B_{f,n}(t) :=  \sup_{j \ge n} \frac{|f^{(j)}(t)|}{e^j M_j}.
\)

* $B_{f,n}(t) \le e^{-n}$ for all $t \in [0,1]$.

* $B_{f,n} \ge B_{f,n+1}$, and $f^{(n)}(t_0) = 0$ implies $B_{f,n}(t_0) = B_{f,n+1}(t_0)$.

* For all $k > n$ and all $t, s \in [0,1]$,

\[
B_{f,n}(s) < \max \{B_{f,n}(t) , e^{-k} \}\, e^{e |t-s| \mu_k}.
\]

(1) and (2) are obvious. To see (3)
let $k > n$, $n \le j < k$, and $t, s \in [0,1]$. Then, by Taylor's formula, for some $\xi$ between $t$ and $s$,

\begin{align*}
\frac{|f^{(j)}(s)|}{e^j M_j}
&\le \sum_{i = 0}^{k-j-1} \frac{|f^{(j+i)}(t)|\, |t-s|^i}{e^j M_j\, i!}
+ \frac{ |f^{(k)}(\xi)|\, |t-s|^{k-j}}{e^j M_j\, (k-j)!}  \\
&= \sum_{i = 0}^{k-j-1} \frac{M_{j+i}}{M_j}\frac{|f^{(j+i)}(t)| }{e^{j+i} M_{j+i}} \frac{(e|t-s|)^i}{i!}
+ e^{-k} \frac{M_{k}}{M_j} \frac{ |f^{(k)}(\xi)|}{M_k} \frac{ (e|t-s|)^{k-j}}{(k-j)!} \\
&\le B_{f,N}(t) \sum_{i = 0}^{k-j-1} \mu_k^i \frac{(e|t-s|)^i}{i!}
+ e^{-k} \mu_k^{k-j}  \frac{ (e|t-s|)^{k-j}}{(k-j)!} \\
&< \max\{ B_{f,n}(t) , e^{-k} \}\, e^{e |t-s| \mu_k},
\end{align*}

where we used that $\mu_k$ is increasing.
If $j \ge k$, then trivially

\begin{align*}
\frac{|f^{(j)}(s)|}{e^j M_j} \le e^{-j} < \max\{ B_{f,n}(t), e^{-k} \} \, e^{e |t-s| \mu_k}.
\end{align*}

This implies (3).

Let $f$ and $x_j$ be as in the theorem.
Set $\ta_k := \sum_{j=1}^{k} |x_{j-1}-x_{j}|$, $k \ge 1$, $\ta_0:= 0$, and define for $t \in [\ta_{n-1},\ta_n]$,

\[
\tilde B_{f,n} (t) :=
\begin{cases}
B_{f,n}(x_{n-1} +\ta_{n-1} -t) & \text{if } x_n < x_{n-1}, \\
B_{f,n}(x_{n-1} -\ta_{n-1} +t) & \text{if } x_n \ge x_{n-1}.
\end{cases}
\]

The function $\tilde B_{f,n}$ is continuous, by (3), and
$\tilde B_{f,n}(\ta_n) = B_{f,n}(x_n) = B_{f,n+1}(x_n) =\tilde B_{f,n+1}(\ta_n)$, by (2).

\[
\tilde B_{f}(t) := \tilde B_{f,n}(t) \quad \text{ if } t \in [\ta_{n-1},\ta_n],~ n \ge 1.
\]

defines a continuous function on $[0,\ta_m]$.
By (1) and (2), we have $\tilde B_{f}(t) \le e^{-n}$ for all $t \ge \ta_{n-1}$,
in particular, $\tilde B_{f}(t) \le e^{-m}$ for all $t \in  [\ta_{m-1}, \ta_m]$.
Since $f$ does not vanish identically,
\(
\max_{t \in [0,\ta_m]} \tilde B_{f}(t) \ge \tilde B_{f}(0) = B_{f,1}(x_0) = B_{f,0}(x_0) >0.
\)
Hence, the range of $\tilde B_f$ contains all numbers $e^{-k}$ for $\ka < k \le m$, where
\(
e^{-\ka} = B_{f,0}(x_0)
\)
which is equivalent to (<ref>).
So we may choose a strictly increasing sequence $t_k$, $\ka < k \le m$, such that $\tilde B_f(t_k) = e^{-k}$
and $\tilde B_f(t) > e^{-k}$ for all
$t \in (t_{k-1},t_k)$ (recursively, take for $t_k$ the smallest $t \in \tilde B_f^{-1}(e^{-k})$ with $t>t_{k-1}$).
By (3) (applied to each interval in the subdivision of $(t_{k-1},t_k)$ induced by the points $\ta_n$ between $t_{k-1}$ and $t_k$)
we may conclude that

\[
\tilde B_{f}(t_{k-1}) \le \tilde B_{f}(t_k)\, e^{e (t_k-t_{k-1}) \mu_k}.
\]

Since $\tilde B_f(t_k) = e^{-k}$,
that means

\[
t_k-t_{k-1} \ge \frac1{e \mu_k}.
\]

Summing over $k$ we find

\[
t_m \ge \frac1{e} \sum_{\kappa < k\le m} \frac1{\mu_k}.
\]

By the choice of the sequence $t_k$, we have $\ta_k \ge t_k$ which implies (<ref>).

Let $M$ be a quasianalytic weight sequence. Let $f \in C^\infty([0,1])$ satisfy (<ref>).
If $f^{(j)}(0) > 0$ for all $j \in \N$, then $f^{(j)}(x) > 0$ for all $x \in [0,1]$ and $j \in \N$.

Suppose that some
derivative $f^{(j)}$ has a zero $x_{j} \in (0,1]$. By Rolle's theorem, we find a strictly decreasing sequence
$x_j > x_{j+1} > \cdots > 0$, where $f^{(k)}(x_k)=0$ for all $k \ge j$. This contradicts <Ref>.

Let $M$ be a quasianalytic weight sequence. Let $f \in C^\infty([0,1])$ satisfy (<ref>).
The total number of zeros (counted with multiplicities) of $f$ is bounded by

\[
\sup\Big\{m \in \N : \sum_{\ka < k \le m} \frac{1}{\mu_k} \le e\Big\}+1.
\]

Let $z_1 \le z_2 \le \cdots \le z_m$ be the zeros of $f$ in $[0,1]$.
By Rolle's theorem, there exists a sequence of points $z_1 = x_0 \le x_1 \le \cdots \le x_{m-1}$ in $[0,1]$ such that $f^{(j)}(x_j) = 0$ for all $j$.
By (<ref>),

\[
1 \ge \sum_{j=1}^{m-1} (x_j - x_{j-1}) \ge \frac{1}{e} \sum_{\ka < k \le m-1} \frac{1}{\mu_k}
\]

and the statement follows.

§.§ The quasianalytic Borel map is never onto

Let $M$ be a quasianalytic weight sequence such that $m_k^{1/k}\to \infty$.
Then there exist elements in $\La^{(M)}$ that are not contained in $j^\infty_{\{0\}}  \cB^{\{N\}}((-r,r))$ for
any quasianalytic weight sequence $N$ and any $r>0$.

Let $a=(a_j) \in \La^{\{M\}}$ be positive, i.e., $a_j >0$ for all $j$.
Let $N$ be any quasianalytic weight sequence and $r>0$.
We claim that
if there exists $f \in \cB^{\{N\}}((-r,r))$ such that $j^\infty_{\{0\}} f = a$ then $f$ is real analytic.
After rescaling we may assume that $\|f^{(j)}\|_{[0,1]} \le N_j$ for all $j \in \N$.
<Ref> implies
that $f^{(j)}(x) >0$ for all $x \in [0,1]$ and all $j \in \N$.
By Bernstein's theorem (e.g. <cit.>), $f$ extends to an analytic function
on the unit disk in $\C$.
Now it suffices to choose $a=(a_j)$ such that it does not define a real analytic germ, which is possible by the assumption
$m_k^{1/k}\to \infty$.

To see that there exist such $a$ even in $\La^{(M)}$ set $L_k := k!\, \sqrt{m_k}$
and let $\ul L$ be the log-convex minorant of $L$. Thus $\ul L \lhd M$ and $\underline \ell_k^{1/k} \to \infty$
(cf. <Ref>), so that $\ul L$ is a quasianalytic weight sequence for which the argument in the
previous paragraph applies.[It is not hard to see that
$\La^{(M)} = \bigcup \{\La^{\{L\}} : L \text{ positive sequence}, ~L \lhd M, ~\ell_k^{1/k} \to \infty\}$.]

§.§ The impossibility of quasianalytic extension

Let $M$ be a quasianalytic weight sequence with $m_k^{1/k} \to \infty$.
Then $\cB^{(M)}((-1,1))$ contains functions $f$ which have no quasianalytic extension to a larger interval, i.e.,
if $\tilde f$ is an extension of $f$ to a neighborhood of $[-1,1]$ and $N$ is any quasianalytic weight sequence, then $\tilde f \not\in \cE^{\{N\}}$.

We use an argument of [65] to see that $\cB^{\{M\}}((-1,1))$ contains
functions with the asserted properties. That such functions can even be found in $\cB^{(M)}((-1,1))$
follows from the reasoning at the end of the proof of <Ref>.
Let $c_j$ be a positive sequence

\begin{gather}
c_j^{1/j} \to 1, \label{eq:cond1} \\
\sum_{j=1}^\infty j^n c_j \le M_n \quad \text{ for all } n \in \N.	\label{eq:cond2}
\end{gather}

The even function
\(
f(x) = \sum_{k=0}^\infty c_{2k} x^{2k}
\)
is real analytic on $(-1,1)$, by (<ref>),
and belongs to $\cB^{\{M\}}((-1,1))$, by (<ref>).
Suppose that $f$ has a quasianalytic extension $\tilde f$ to a larger interval.
Then $\tilde f^{(j)} (0) =  f^{(j)} (0) \ge 0$ for all $j \in \N$.
Hence $\tilde f$ is analytic on a disk centered at $0$ with radius larger than $1$, by <Ref> und Bernstein's theorem.
But this contradicts (<ref>).

§.§ Digression: intricacies of the quasianalytic setting

We want to point out some interesting intricacies of the quasianalytic setting. This section deviates from
the central theme of the article it is thus kept rather short.

(1) For each smooth germ $f$ there exist quasianalytic weight sequences $M^1,M^2$ such that
$f =f_1 + f_2$ with $f_i \in \cE^{\{M^i\}}$, $i=1,2$. See [57] and
[80] (who concluded from this that there is no largest o-minimal expansion of the real field).

(2) Weierstrass division and preparation, generally, fail in quasianalytic classes.
See [2, 35, 67].
If $M$ is derivation-closed and quasianalytic, then division in $\cE^{\{M\}}$ by a Weierstrass polynomial $\vh$ holds
if and only if all roots of $\vh$ are real; see [30] and [29].
For an overview of related results see [88].

(3) Assume that $M$ is a quasianalytic strongly log-convex and derivation-closed weight sequence.
It is not known if the local ring $\cE^{[M]}_0$ is Noetherian.
But $\cE^{[M]}$ admits resolution of singularities [8, 9] and also [80]
and, consequently, tools such as
topological Noetherianity, curve selection, and Łojasiewicz inequalities are available.

(4) Ultradifferentiable quasianalyticity cannot be tested in lower dimensions (in contrast to real analyticity
[86, 11, 12]).
Let $M$ be a quasianalytic strongly log-convex weight sequence with $m_k^{1/k}\to \infty$.
For any $n\ge 2$ and any positive sequence $N$
there exists a $C^\infty$-function $f \in \cE^{[M]}(\R^n \setminus \{0\}) \setminus \cE^{\{N\}}(\R^n)$
such that $f \o p \in \cE^{[M]}$ for all $\cE^{[M]}$-mappings $p : \R^m \supseteq U \to \R^n$ with $m<n$.
See [45] and [72].

(5) Quasianalytic Roumieu classes are intersections of non-quasianalytic ones:
Let $M$ be a quasianalytic strongly log-convex weight sequence. Then

\[
\cE^{\{M\}} = \bigcap_{N \in \cN(M)} \cE^{\{N\}},
\]

where $\cN(M)$ is the collection of all  non-quasianalytic weight sequences $N \ge M$.
For suitable $M$ (e.g. Denjoy sequences $Q^{n,1}$, see <Ref>), we may even restrict the intersection to all
strongly log-convex weight sequences in $\cN(M)$. But notice that we cannot describe all $\cE^{\{M\}}$
in this way: the intersection of all non-quasianalytic $\cE^{\{N\}}$, where $N$ is strongly log-convex,
is the quasianalytic class $\cE^{\{Q^{1,1}\}} \supsetneq C^\om$.
See [13] and [49] as well as [66] for an application.

(6) Let $M$ be a quasianalytic, strongly log-convex,  and derivation-closed weight sequence.
Consider the quasianalytic equation

\[
\vh(x,y) = y^d + a_1(x) y^{d-1} + \cdots a_d(x) \in \cE^{[M]}_0[y], \quad x=(x_1,\ldots,x_n).
\]

Due to [89], if $n = 1$, then a smooth solution $y=h(x)$ is of class $\cE^{[M]}$.
It is not known, if this is true for $n>1$.
There is the following partial solution [6]:
Let $\vh(x,y)$ be a function of class $\cE^{\{M\}}$ defined near $(a,b) \in \R^n \times \R$.
Then there exist $p \in \N$ and a quasianalytic class $\cQ \subseteq \cE^{\{M^{(p)}\}}$ (where $M^{(p)}_k := M_{pk}$)
such that if $\vh(x,y)=0$ admits a formal power series solution $y = H(x)$ at $a$ then
there is a solution $y = h(x) \in \cQ$ near $a$ and $T_a h = H$. Note that, in general,
$M^{(p)}$ is no longer quasianalytic and $\cQ$ is not a Denjoy–Carleman class.

§ BOREL'S LEMMA WITH CONTROLLED LOSS OF REGULARITY

Let $M$ be a weight sequence satisfying $m_k^{1/k} \to \infty$.
We have seen that the Borel map $j^\infty_{\{0\}} : \cE^{[M]}(\R) \to \La^{[M]}$
is injective if and only if $M$ is quasianalytic and
surjective if and only if $M$ is strongly non-quasianalytic.
If $M$ is not strongly non-quasianalytic, then several new natural questions arise.

Assume that $M$ is not strongly non-quasianalytic.

* What can be said about the image $j^\infty_{\{0\}} \cE^{[M]}(\R)$? Is there an intrinsic description?

* In the case that $M$ is quasianalytic, is there a constructive method for finding the unique function $f$
with $j^\infty_{\{0\}}f=a$ for a given $a \in j^\infty_{\{0\}} \cE^{[M]}_0$?

* When do we have $\La^{[L]} \subseteq j^\infty_{\{0\}} \cE^{[M]}(\R)$ for another weight sequence $L$?

Note that, as see in <Ref>,
positive sequences that grow fast enough cannot belong to $j^\infty_{\{0\}} \cE^{[M]}_0$ if
$M$ is quasianalytic.

§.§ When do we have $\La^{[L]} \subseteq j^\infty_{\{0\}} \cE^{[M]}(\R)$?

This question was completely answered by Schmets and Valdivia [85]
if $M$ is a non-quasianalytic weight sequence.

Let $M\le N$ be weight sequences, $N$ non-quasianalytic.
For $p \in \N_{\ge 1}$ we consider the sequence $\la_p = \la_p(M,N)$ defined by

\[
\la_{p,k} := \sup_{0 \le j <k} \Big(\frac{M_k}{p^k N_j} \Big)^{\frac{1}{k-j}},\quad  k \ge 1.
\]

Note that $\la_p \le \mu$ for all $p\ge 1$, indeed, since $M \le N$,

\[
\Big(\frac{M_k}{p^k N_j} \Big)^{\frac{1}{k-j}}
\le \Big(\frac{M_k}{M_j} \Big)^{\frac{1}{k-j}}
= (\mu_{j+1} \cdots \mu_k)^{\frac{1}{k-j}} \le \mu_k.
\]

If $M$ has moderate growth, then $\mu_k \lesssim M_k^{1/k}$ and so also a converse estimate holds:

\begin{align*}
\mu_k \lesssim M_k^{1/k} =  p \Big(\frac{M_k}{p^k N_0}\Big)^{1/k} \le  p \la_{p,k}.
\end{align*}

Let $M\le N$ be weight sequences, $N$ non-quasianalytic.
The following conditions are equivalent:

* $\La^{(M)} \subseteq j^\infty_{\{0\}} \cE^{(N)}(\R)$.

* $\La^{\{M\}} \subseteq j^\infty_{\{0\}} \cE^{\{N\}}(\R)$.

* There is $p \ge 1$ such that

\begin{equation} \label{eq:SVcond}
\sup_{k \ge 1} \frac{\la_{p,k}}{k} \sum_{j \ge k} \frac{1}{\nu_j} < \infty.
\end{equation}

By the remarks before the theorem, if $M$ has moderate growth, then (<ref>) (for any $p \ge 1$) is equivalent to

\begin{equation}\label{eq:mixedga1}
\sup_{k \ge 1} \frac{\mu_k}{k} \sum_{j \ge k} \frac{1}{\nu_j} < \infty.
\end{equation}

In general, the latter condition is stronger than (<ref>).
By <Ref>, the descendant $\si$ of $\nu$ is the largest sequence among all sequences $\mu$
satisfying $\mu \lesssim \nu$ and (<ref>), in the sense that it generates the largest function space.

§.§ Moment-type summation

Some interesting results concerning the questions <ref>(1) and (2)
based on a moment-type summation method
are due to
Beurling, Carleson [25] and recently Kiro [46].

Informally, it works as follows. Let $I \subseteq \R$ be an open interval containing $0$.
Given $f \in \cE^{(M)}(I)$, the singular transform of $f$ is the entire function

\[
S_M f (z) := \sum_{n \ge 0} \frac{f^{(n)}(0)}{n!\, m_{n+1}} z^n.
\]

It depends only on $j^\infty_{\{0\}} f$
and hence studying the image $S_M \cE^{(M)}(I)$ is equivalent to studying the image of the Borel map
This study is based on the observation that elements in $S_M \cE^{(M)}(I)$ have a certain growth behavior in
horizontal strips.
(In the Roumieu case the series may have a finite radius of convergence and analytic continuation to a horizontal strip
is required.)

Let $K$ be the kernel that solves the moment problem

\[
\int_0^\infty t^n K(t) \, dt = m_{n+1}, \quad n \in \mathbb N.
\]

Under suitable conditions, the regular transform

\[
R_M F(x) := \int_0^\infty F(xt) K(t)\, dt
\]

then yields the desired reconstruction $R_M S_M f =f$.

To make this approach work, one needs to control the asymptotic behavior of $K$ and of

\[
E(z) := \sum_{n \ge 0} \frac{z^n}{m_{n+1}}
\]

which manifests itself in regularity properties of the weights $M$. The function $E$ regulates the growth
behavior of elements in $S_M \cE^{(M)}(I)$ in horizontal strips.

Kiro [46] executes this program for a suitable class of weights; the case $M_k = k! \log(k +e)^k$
was treated by Beurling. Also a duality between suitable non-quasianalytic and quasianalytic classes, which have the
same image under the respective singular transform, is explored.

§ OPTIMAL CUTOFF FUNCTIONS

The most important tool for the extension problem for general closed subsets $A \subseteq \R^n$ are good cutoff functions.
We start with an observation that shows how good ultradifferentiable cutoff functions can be.

§.§ A lower bound for cutoff functions

Cf. [20].
Let $M=(M_k)$ be a non-quasianalytic weight sequence.
Let $r,\ep,\rh>0$.
Suppose that $\vh \in \cB^{M}_\rh(\R)$ satisfies

\begin{equation} \label{eq:Bruna1}
\vh|_{[-r,r]} = 1 \quad \text{ and } \quad
\on{supp} \vh \subseteq [-(1+\ep)r,(1+\ep)r].
\end{equation}

It is to be expected that $\|\vh\|^M_{\R,\rh}$ tends to infinity,
if $r$, $\ep$, or $\rh$ approach $0$.
Let us check and quantify this guess.
For $t \in (r,(1+\ep)r)$, we have by Taylor's formula

\[
|\vh(t)| \le \frac{\vh^{(k)}(\ta)}{k!} (r \ep)^k, \quad k \in \N,
\]

for some $\ta \in (t,(1+\ep)r)$. Since $\vh(t)\to 1$ as $t \to r$, we find
$\|\vh^{(k)}\|_\R \ge k!/(r\ep)^k$ for all $k$. Thus,

\[
\|\vh\|^M_{\R,\rh} \ge \sup_{k \in \N} \frac{k!}{(\rh r \ep)^k M_k} = \frac{1}{\inf_{k \in \N} (\rh r \ep)^k m_k}.
\]

It turns out that $h_m(t) := \inf_{k \in \N} m_k t^k$ is a very useful auxiliary function; we will discuss it in <Ref>.
So any cutoff function $\vh \in \cB^{M}_\rh(\R)$ with (<ref>)
must satisfy

\begin{equation} \label{eq:goodcutoff}
\|\vh\|^M_{\R,\rh} \ge  \frac{1}{h_m(\rh r \ep)}.
\end{equation}

§.§ Associated functions

Suppose that $m=(m_k)$ is log-convex, $1 = m_0 \le m_1$ and $m_k^{1/k} \to \infty$
(i.e., $m$ is a weight sequence or, equivalently, $M_k = k!\, m_k$ is a
strongly log-convex weight sequence).
We associate the function $h_m : [0,\infty) \to [0,\infty)$, where  $h_m(0):=0$ and

\begin{equation} \label{h}
h_m(t) := \inf_{k \in \N} m_k t^k, \quad t > 0.
\end{equation}

Then $h_m$ is increasing, continuous, and positive for $t>0$. For $t \ge 1/m_1$ we have $h_m(t) = 1$. From $h_m$
we may recover the sequence $m$ by $m_k = \sup_{t>0} h_m(t)/ t^{k}$.
We also associate the counting function $\Ga_m : (0,\infty) \to \N_{\ge 1}$ by setting

\begin{equation} \label{counting2}
\Ga_m(t) := \min\{k \ge 1 : h_m(t) =  m_k t^k\} = \min\Big\{k \ge 1 : \frac{m_{k+1}}{m_k} \ge \frac{1}{t} \Big\};
\end{equation}

for this identity we need that $\frac{m_{k+1}}{m_k}$ is increasing, i.e., $m$ is log-convex. Then:

\begin{gather}
\text{$k \mapsto m_k t^k$ is decreasing for $k \le \Ga_m(t)$,} \label{GaProp1}
\\
\text{$h_m(t) = m_{\Ga_m(t)} t^{\Ga_m(t)} \le m_k t^k$ for all $k$.} \label{GaProp2}
\end{gather}

It follows that

\begin{equation} \label{GaProp3}
\Ga_m(t) = k \quad \text{ and }\quad h_m(t) = m_k t^k, \qquad \text{ for } t \in [\tfrac{m_k}{m_{k+1}},\tfrac{m_{k-1}}{m_{k}}).
\end{equation}

For $m_k = k!^s$, where $s > 0$, the function $h_m$ behaves like $t \mapsto \exp(-1/t^{1/s})$:

\begin{align*}
\sup_{k} \frac{t^k}{k!^s} = \Big( \sup_{k}  \frac{t^{k/s}}{k!} \Big)^s \le \Big( \sum_{k}  \frac{t^{k/s}}{k!} \Big)^s = e^{s t^{1/s}}
\end{align*}

and, conversely,

\begin{align*}
e^{st^{1/s}} = \Big(\sum_{k} \frac{1}{2^k}  \frac{(2 t^{1/s})^k}{k!} \Big)^s \le 2^s \sup_{k} \frac{(2^s t)^k}{k!^s}.
\end{align*}

Similarly, for $m_k = \log(k+e)^{sk}$,  $h_m$ behaves like $t \mapsto \exp(-\exp(1/t^{1/s}))$.

We shall need some properties of $h_m$ and $\Ga_m$ which are guaranteed if
$m$ has moderate growth:

Suppose that $m$ is a weight sequence of moderate growth, i.e.,
there is a constant $C>1$ such that

\begin{equation} \label{eq:mgv1}
\frac{m_{2k}}{m_{2k-1}} \le C \frac{m_{k}}{m_{k-1}},  \quad k\ge 1.
\end{equation}

Then[Actually (<ref>) and (<ref>) are equivalent, see <cit.>.]

\begin{gather}
h_m(t) \le h_m(Ct)^2, \quad t>0, \label{hmg}
\end{gather}

and, for a possibly other constant $C>1$,

\begin{gather}
2  \Ga_m(Ct) \le \Ga_m(t),  \quad t>0 \text{ small enough}.   \label{eq32}
\end{gather}

Consider the function
\(
\Si_m(t) := |\{k \ge 1 : \tfrac{m_k}{m_{k-1}} \le t\}|.
\)
Now (<ref>) implies

\begin{equation} \label{eq:Sigmam}
2 \Si_{m}(t) \le \Si_m (Ct), \quad t>0.
\end{equation}

It is well-known (cf. [58] and [47]) that $\om_m$ defined by
$\om_m(t) = \log(1/h_m(1/t))$
satisfies[Note that the integral
$\int_1^\infty \frac{\om_M(t)}{t^2}\,dt$ converges if and only if $\int_1^\infty \frac{\Si_M(t)}{t^2}\,dt$ converges, and that is the case if and only if $M$ is
non-quasianalytic; <cit.>.]

\begin{equation} \label{omM}
\om_m(t) = \int_{0}^t\frac{\Si_m(u)}{u} \,du.
\end{equation}

So (<ref>) implies $2\omega_{m}(t) \le \omega_m(Ct)$ for all $t>0$, which is clearly equivalent to (<ref>).

Let us check (<ref>). Since $m$ has moderate growth and is log-convex, there are constants $C_1,C_2>0$ such that

\[
\frac{m_{2k+1}}{m_{2k}} \le C_1 \frac{m_{2k}}{m_{2k-1}} \le C_2 \frac{m_{k}}{m_{k-1}} \le C_2 \frac{m_{k+1}}{m_{k}},  \quad k\ge 1.
\]

Fix $t>0$ and let $\ell := \Ga_m(t)$. If $\ell=2k$,

\[
\frac{1}{t} \le \frac{m_{\ell+1}}{m_{\ell}} \implies \frac{1}{C_2 t} \le    \frac{m_{k+1}}{m_{k}},
\]

that is, $\Ga_m(C_2t) \le k = \tfrac{1}{2} \Ga_m(t)$.
If $\ell = 2k+1$, then we similarly see that
$\Ga_m(\frac{C_2}{C_1}t) \le k =  \tfrac{1}{2} (\Ga_m(t) -1) \le \tfrac{1}{2} \Ga_m(t)$.
Since $\Ga_m$ is decreasing, (<ref>) follows.

Moderate growth of the weight sequence is an essential technical tool in this section. So we make the following definition.

A weight sequence $M=(M_k)$ is called strongly regular if $M$ is strongly non-quasianalytic, has moderate growth,
and $m$ is log-convex.
Note that the last condition is actually for free:
a strongly non-quasianalytic weight sequence $M$ is equivalent to a strongly log-convex weight sequence (that is still
strongly non-quasianalytic),
by <Ref>.

§.§ Optimal cutoff functions of Roumieu type

It turns out that there exist optimal $\cE^{\{M\}}$ cutoff functions, i.e., $\vh \in \cE^{\{M\}}(\R)$ satisfying (<ref>) and realizing (<ref>) so that

\begin{equation*}
\|\vh\|^M_{\R,\rh} \sim  \frac{1}{h_m(\rh r \ep)},
\end{equation*}

if and only if $M$ is strongly non-quasianalytic.
The existence of $\vh$ will follow from <Ref>. The following lemma provides the necessary assumptions.
Recall that $\mu_k^* = \frac{\mu_k}{k} = \frac{m_k}{m_{k-1}}$.

Let $M$ be a strongly regular weight sequence.
There is a constant $A\ge 1$ such that for each integer $p\ge 1$ there is a sequence $(\alpha^p_k)_{k\in\N}$ satisfying

\begin{align}
\sum_{k\ge 0}\frac{\alpha^p_k}{\alpha^p_{k+1}}\le 1, \quad \alpha_0^p=1,
\quad 0<\alpha^p_k\le   \frac{(\frac{A}{\mu_{p+1}^*})^k M_k}{h_{m}(\frac{1}{e\mu_{p}^*})}.
\label{alpha3}
\end{align}

We will show that, for a suitable constant $A \ge 1$, the family of sequences

\begin{equation*}
\al^p_k :=
\begin{cases}
\big(\frac{A}{\mu_{p+1}^*}\big)^k M_k & \text{ if } k >p, \\
(2p)^k & \text{ if } k \le p,
\end{cases}
\end{equation*}

has the desired properties.
Since $M$ has moderate growth, $\mu_{k+1} \lesssim  M_k^{1/k}$ and thus,
for some constant $C \ge 1$,

\begin{align*}
\alpha^p_p= 2^pp^p\le  \Big( \frac{2C}{\mu_{p+1}^*}\Big)^p M_p.
\end{align*}

So if $A$ is chosen large enough, then

\begin{align*}
\sum_{k\ge 0} \frac{\alpha_k^p}{\alpha^p_{k+1}}
&\le\sum_{k<p}\frac{1}{2p}+ \frac{\mu_{p+1}^*}{A}\sum_{k\ge p}\frac{1}{\mu_{k+1}}
\le 1,
\end{align*}

since $M$ is strongly non-quasianalytic.
The bound for $\al^p_k$ (in (<ref>)) is obvious for $k>p$,
since $h_m\le 1$.
If $k\le p$, then, for large enough $A$,

\begin{align*}
\frac{\al^p_k}{(\frac{A}{\mu_{p+1}^*})^k\,  M_k}
&=\frac{2^k p^k}{(\frac{A}{\mu_{p+1}^*})^k\,  M_k}
\le \frac{\mu_{p+1}^k}{(\frac{A}{2})^{k} M_k}
\le \frac{\mu_{p}^k}{M_k}
\le \frac{\mu_{p}^p}{M_p} = \frac{\mu_{p}^p}{p! m_p} \le  \frac{(e \mu_p^*)^p}{m_p}.
\end{align*}

This implies the statement.

Let $M=(M_k)$ be a non-quasianalytic strongly log-convex weight sequence of moderate growth.
Then the following conditions are equivalent:

* $M$ is strongly non-quasianalytic.

* There is a constant $C>0$ such that for all $\rh, r, \ep>0$ there exists $\vh \in \cE^{\{M\}}(\R)$ satisfying (<ref>) and

\begin{equation} \label{eq:cutoffRM}
\|\vh\|^M_{\R,\rh} \le  \frac{1}{h_m(C \rh r \ep)}.
\end{equation}

(1) $\Rightarrow$ (2)
We first consider the case $r=\ep=1$.
Let $A$ be the constant from <Ref>.
Fix $0<\eta\le 2 A$.
Since $\mu_k^* \nearrow \infty$,
there is an integer $p \ge 1$ such that

\begin{equation}\label{testfunctionconstructionequ1}
\frac{2A}{\mu_{p+1}^*} < \et \le  \frac{2A}{\mu_p^*}.
\end{equation}

By <Ref>, we may
apply <Ref> (to $d_{k+1}=\frac{\alpha^p_k}{\alpha^p_{k+1}}$ and $K =[-1,1]$) and
get $\vh =\vh_{\et} \in C^\infty(\R)$ with
$0 \le \vh \le 1$, $\vh|_{[-1,1]} = 1$, $\on{supp}\vh \subseteq [-2,2]$, and

\begin{align*}
\le 2^{k}\alpha_k^p
\le \frac{\eta^k M_k}{h_m(\frac{\et}{6A})}, \quad k \ge 1,
\end{align*}

by (<ref>) and (<ref>).
For $\eta>2A$ we put $\vh_{\eta}:=\vh_{2A}$; then since $h_m \le1$,

\begin{align*}
|\vh^{(k)}(t)| &\le \frac{(2A)^k M_k}{h_m(\frac{2A}{6A})}
\le \frac{1}{h_m(\frac{1}{3})}  \frac{\eta^k M_k}{h_m(\frac{\et}{6A})}.
\end{align*}

If $\de:= 1/h_m(\frac{1}{3})$ then for every $\eta>0$,

\begin{equation*}
|\vh^{(k)}(t)| \le \frac{(\de \eta)^k M_k}{h_m(\frac{\et}{6A})}.
\end{equation*}

This implies (2), for $r=\ep=1$.
For the general case it suffices to compose $\vh_{\rh r \ep}$
with an
odd smooth function $\th : \R \to \R$ satisfying $\th([-r,r]) = [-1,1]$ and
$\th(x) = \frac{x + r\ep -r}{r\ep}$ for $x \ge r$.

(2) $\Rightarrow$ (1)
Let $\vh$ be as in (2) for given $r>0$ and $\rh = \ep =1$.
Then the function $\ps(x):= \vh(x-2r) \cdot h_m(Cr)$ satisfies $\|\ps^{(k)}\|_\R \le M_k$ for all $k$ and vanishes on $(-\infty,0]$.
<Ref> implies

\[
h_m(Cr) = |\ps(r)| \le \Big( \frac{2r}{\sum_{k \ge \ell} \frac{1}{\mu_k}} \Big)^\ell, \quad \ell \ge 1,
\]

provided that $r< \frac{1}{4}\sum_{k \ge \ell} \frac{1}{\mu_k}$.
For those $\ell$ that satisfy $\frac{1}{C \mu_\ell^*} < \frac{1}{4}\sum_{k \ge \ell} \frac{1}{\mu_k}$,
we obtain

\[
h_m(\tfrac{1}{\mu^*_\ell}) \le \Big( \frac{\tfrac{2}{C}\tfrac{1}{\mu^*_\ell}}{\sum_{k \ge \ell} \frac{1}{\mu_k}} \Big)^\ell.
\]

On the other hand,

\[
h_m(\tfrac{1}{\mu^*_\ell}) \stackrel{\eqref{GaProp3}}{=} m_{\ell-1} (\tfrac{1}{\mu^*_\ell})^{\ell-1} \ge H^{\ell-1}
\]

for some constant $H>0$, since $m$ has moderate growth. So we may conclude that $\sum_{k \ge \ell} \frac{1}{\mu_k} \lesssim \frac{1}{\mu^*_\ell}$
for all $\ell$.

§.§ Optimal cutoff functions of Beurling type

There exist also optimal cutoff functions of Beurling type. They will play a crucial role in <Ref>.

Let $M$ be a strongly regular weight sequence.
Then there exist functions $(\vh_r)_{r>0}$ in $\cE^{(M)}(\R)$ with the following properties:

* $0 \le \vh_r \le 1$ for all $r>0$.

* $\vh_r|_{[-r,r]} = 1$ and $\supp \vh_r \subseteq [- \frac{9}8r,\frac{9}8r]$ for all $r>0$.

* For each $\rh>0$ there exist constants $A_\rh>0$ and $b_\rh>0$, such that

\begin{equation} \label{eq:cutoffBM}
\|\vh_r\|^M_{\R,\rh} \le  \frac{A_\rh}{h_m(b_\rh r)}, \quad r>0.
\end{equation}

The essential difference to <Ref> is that the cutoff functions may be taken independent of $\rh$.
The dependence of the argument of $h_m$ on $\rh$ in (<ref>) is however simpler than in (<ref>);
the linear dependence in (<ref>) will be used in the proof of <Ref>.

We may assume the setup of <Ref>, in particular,
there is a constant $A>0$ such that (<ref>) holds.
Let $0<r \le 8A$ be fixed and take an integer $k=k(r) \ge 1$ such that

\[
\frac{8A}{\mu_{k}^*} < r \le \frac{8A}{\mu_{k-1}^*}.
\]

By <Ref> (applied to $K=[-r,r]$ and $\sum d_j$ the left-hand side of (<ref>)),
there exists $\vh=\vh_r \in C^\infty(\R)$
$0 \le \vh \le 1$, $\vh|_{[-r,r]} = 1$, $\on{supp} \vh \subseteq [-\frac{9}8r,\frac{9}8r]$, and

\begin{align*}
|\vh_r^{(j)}| \le
\begin{cases}
2^j \mu_k^j & \text{ if } 0 \le j \le k,
\\
2^j \mu_k^k \frac{M_j}{M_k} \Big( \frac{k^{j-k} k!}{j!}\Big)^\ep & \text{ if } j>k.
\end{cases}
\end{align*}

Since $m$ has moderate growth, we have $\mu^*_k \le C \mu^*_{k-1}$ for some $C \ge 1$. Thus

\begin{equation*}
\frac{\mu_k^k}{M_k} = \frac{\mu_k^k}{k!\,m_k} \le \frac{(e\mu_k^*)^k}{m_k} \le \frac{(Ce\mu_{k-1}^*)^k}{m_k}
\le \Big(\frac{8ACe}{r}\Big)^k \frac{1}{m_k}.
\end{equation*}

Let $\rh \in (0,1)$ be given.
If $j \le k$, then $\mu_k^j \le \mu_k^k \frac{M_j}{M_k}$, and hence

\begin{align*}
|\vh_r^{(j)}| \le 2^j M_j \Big(\frac{8ACe}{r}\Big)^k \frac{1}{m_k}
\le \rh^j M_j \Big(\frac{16ACe}{r\rh}\Big)^k \frac{1}{m_k}.
\end{align*}

If $j> k$, then $\frac{k^{j-k} k!}{j!} \le \frac{k^{j-k}}{(j-k)!}$, and so

\begin{align*}
|\vh_r^{(j)}| &\le 2^j M_j \Big(\frac{8ACe}{r}\Big)^k \frac{1}{m_k}  \Big( \frac{k^{j-k} }{(j-k)!}\Big)^\ep
\\
&= \rh^j M_j \Big(\frac{16ACe}{r \rh}\Big)^k \frac{1}{m_k}  \Big( \frac{(k(\frac{2}{\rh})^{1/\ep})^{j-k} }{(j-k)!}\Big)^\ep
\\
&\le \rh^j M_j \Big(\frac{16ACe}{r \rh}\Big)^k \frac{1}{m_k} e^{\ep k (\frac{2}{\rh})^{1/\ep}}
\\
&= \rh^j M_j  \frac{1}{(b_\rh r)^k m_k}, \quad \text{ where }\quad b_\rh := \frac{\rh}{16ACe^{1+ \ep (\frac{2}{\rh})^{1/\ep}}}.
\end{align*}

It follows that, for all $0 < r \le 8A$ and all $\rh >0$,

\begin{align*}
\|\vh_r\|^M_{\R,\rh} \le  \frac{1}{h_m(b_\rh r)};
\end{align*}

for $\rh\ge 1$ the estimates are much simpler.
For $r> 8A$ we apply <Ref> to $K=[-r,r]$ and (<ref>) with $k=1$.
This gives $\vh_r$ having the same bounds as $\vh_{8A}$. Then

\[
\|\vh_r\|^M_{\R,\rh} \le  \frac{1}{h_m(b_\rh 8A)} \le  \frac{1}{h_m(b_\rh 8A)}  \frac{1}{h_m(b_\rh r)} =: \frac{A_\rh}{h_m(b_\rh r)},
\]

since $h_m \le 1$.

§.§ Whitney cubes

We denote by $d_A(z) =d(z,A) = \inf\{|z-y| : y \in A\}$ the Euclidean distance of $z \in \R^n$ to some set $A \subseteq \R^n$.

For each closed non-empty set $A \subseteq \R^n$ there exists a countable collection of closed cubes $(Q_j)_{j \ge 1}$ with sides parallel to the coordinate
axes and the following properties.

* $\R^n \setminus A = \bigcup_{j\ge 1} Q_j$.

* The interiors of the cubes $Q_j$ are pairwise disjoint.

* For each $j \ge 1$ we have
$$\on{diam} Q_j \le \on{dist} (Q_j,A) \le 4 \on{diam} Q_j.$$

* For each $j\ge 1$ let $Q_j^*$ be image of $Q_j$ under the dilation by the factor $\tfrac{9}{8}$ with center the center of $Q_j$.
There exist constants $0< b_0 \le 1 \le B_0$ such that for all $j \ge 1$ and all $z \in Q_j^*$,

\[
b_0 \on{diam} Q_j \le d_A(z) \le B_0 \on{diam} Q_j.
\]

* For each $j\ge 1$ the number of $i \ge 1$ with $Q_i^* \cap Q_j^* \ne \emptyset$ is bounded by $12^{2n}$.

* There exist constants $0< b_1 \le 1 \le B_1$ such that for all $i,j\ge 0$ with $Q_i^* \cap Q_j^* \ne \emptyset$ we have

\[
b_1 \on{diam} Q_j \le \on{diam} Q_i \le B_1 \on{diam} Q_j.
\]

The constants $b_0,B_0,b_1,B_1$ are independent of $A$.

A collection $(Q_j)_{j\ge 1}$ with these properties is called a family of Whitney cubes for $A$.

§.§ Optimal partitions of unity

Having optimal cutoff functions, it is a standard procedure to construct
optimal partitions of unity subordinate to a given family of Whitney cubes.
We begin with the Roumieu case.

Let $M$ be a strongly regular weight sequence.
Let $A \subseteq \R^n$ be a non-empty closed set and $(Q_j)_{j \ge 1}$ a family of Whitney cubes for $A$.
Then there exists
$C_1\ge 1$ such that for all $\ep>0$ there is a family of $C^\infty$-functions
$(\vh_{j,\ep})_{j \ge 1}$ satisfying

* $0\le\vh_{j,\ep}\le 1$ for all $j \ge 1$,

* $\on{supp}\vh_{j,\ep}\subseteq Q^*_j$ for all $j\ge 1$,

* $\sum_{j\ge 1}\vh_{j,\ep}(x)=1$ for all $x \in \R^n\setminus A$,

* for all $j\ge 1$, $\be \in\N^n$, and $x\in\R^n\setminus A$,

\begin{equation*}
|\vh^{(\be)}_{j,\ep}(x)| \le \frac{\ep^{|\be|} M_{|\be|}}{h_m(C_1\ep\, d_A(x))}.
\end{equation*}

By <Ref>, there is a constant $C >1$ such that for each $\rh>0$ and each $r>0$
there exist functions $\ch_{\rh,r}$ such that

* $0\le\ch_{\rh,r}(x)\le 1$ for all $x\in\R^n$,

* $\ch_{\rh,r}(x)=1$ for all $x\in [-r,r]^n$,

* $\ch_{\rh,r}(x)=0$ for all $x\not\in [-\frac{9}{8}r,\frac{9}{8}r]^n$,

* for all $\be \in\N^n$ and all $x\in\R^n$,

\begin{equation*}
|\ch^{(\be)}_{\rh,r}(x)|\le \frac{\rh^{|\be|} M_{|\be|}}{h_m(C\rh r)}.
\end{equation*}

Let $x_j$ be the center and $2 s_j$ the sidelength of $Q_j$. Set

\[
\ps_{j,\ep}(x) := \ch_{\frac{\ep}{12^{2n}},s_j}(x-x_j).
\]

Then $\ps_{j,\ep}$ is $1$ on $Q_j$ and $0$ outside $Q_j^*$.

\begin{align*}
|\ps^{(\be)}_{j,\ep}(x)| \le \frac{(\frac{\ep}{12^{2n}})^{|\be|} M_{|\be|}}{h_{m}(  \frac{C\ep s_j}{12^{2n}} )}
\le \frac{(\frac{\ep}{12^{2n}})^{|\be|} M_{|\be|}}{h_{m}(C_0 \ep\, d_A(x))}, \quad \text{ for } C_0 := \frac{C}{2 \sqrt{n} 12^{2n} B_0},
\end{align*}

since $d_A(x) \le B_0 \on{diam} Q_j = 2 \sqrt{n} B_0 s_j$ if $x \in Q^*_j$, by <Ref>(4).
Now define $(\vh_{j,\ep})_{j \ge 1}$ by

\begin{equation} \label{eq:defpartition}
\vh_{1,\ep}:=\psi_{1,\ep}, \quad \vh_{j,\ep}:=\ps_{j,\ep} \prod_{k=1}^{j-1}(1-\psi_{k,\ep}), ~ j\ge 2.
\end{equation}

It is clear that $0\le\vh_{j,\ep}\le 1$ and $\on{supp}\vh_{j,\ep}\subseteq Q^*_j$ for all $j \ge 1$.
To see (3) observe that, for all $j \ge 1$,

\[
\vh_{1,\ep} + \cdots + \vh_{j,\ep} = 1 - (1-\ps_{1,\ep}) \cdots (1-\ps_{j,\ep}).
\]

If $x \in \R^n \setminus A$, then $x \in Q_{i_x}$ for some $i_x\ge 1$ and thus $\ps_{i_x,\ep}(x) =1$.
Since the family $(\supp \vh_{j,\ep})_{j\ge 1}$ is locally finite, it follows that $\sum_{j\ge 1}\vh_{j,\ep}(x)=1$ for all $x \in \R^n\setminus A$.
Finally, let us check (4).
In the product defining $\vh_{j,\ep}$ at most $12^{2n}$ factors are different from $1$, by <Ref>(5).

\[
\le \frac{\ep^{|\be|} M_{|\be|}}{h_{m}(C_0 \ep\, d_A(x))^{12^{2n}}}.
\]

By <Ref>, there is a constant $D>1$ such that $h_m(t) \le h_m(Dt)^{12^{2n}}$ for all $t>0$.
Thus, we obtain (4) with $C_1 = C_0/D$.

Let us come to the Beurling case.

Let $M$ be a strongly regular weight sequence.
Let $A \subseteq \R^n$ be a non-empty closed set and $(Q_j)_{j \ge 1}$ a family of Whitney cubes for $A$.
Then there is a family of $C^\infty$-functions
$(\vh_{j})_{j \ge 1}$ satisfying

* $0\le\vh_{j}\le 1$ for all $j \ge 1$,

* $\on{supp}\vh_{j}\subseteq Q^*_j$ for all $j\ge 1$,

* $\sum_{j\ge 1}\vh_{j}(x)=1$ for all $x \in \R^n\setminus A$,

* for each $\rh>0$ there exist $C_1,\ta_1>0$ such that
for all $j\ge 1$, $\be \in\N^n$, and $x\in\R^n\setminus A$,

\begin{equation*}
|\vh^{(\be)}_{j}(x)| \le C_1 \frac{\rh^{|\be|} M_{|\be|}}{h_m(\ta_1\, d_A(x))}.
\end{equation*}

By <Ref>, there exist smooth functions $(\ch_r)_{r>0}$ such that

* $0\le\ch_{r}(x)\le 1$ for all $x\in\R^n$,

* $\ch_{r}(x)=1$ for all $x\in [-r,r]^n$,

* $\ch_{r}(x)=0$ for all $x\not\in [-\frac{9}{8}r,\frac{9}{8}r]^n$,

* for each $\rh>0$ there exist $A_\rh\ge 1$ and $b_\rh>0$ such that for all $\be \in\N^n$ and $x\in\R^n$,

\begin{equation*}
|\ch^{(\be)}_{r}(x)|\le A_\rh \frac{\rh^{|\be|} M_{|\be|}}{h_m(b_\rh r)}.
\end{equation*}

If $x_j$ denotes the center of the cube $Q_j$ and $2 s_j$ its sidelength, we set

\[
\ps_{j}(x) := \ch_{s_j}(x-x_j).
\]

$\ps_{j}|_{Q_j} =1$, $\on{supp} \ps_j \subseteq Q_j^*$, and

\begin{align*}
|\ps^{(\be)}_{j}(x)| \le  A_\rh \frac{\rh^{|\be|} M_{|\be|}}{h_{m}( b_\rh s_j)}
\le A_\rh \frac{\rh^{|\be|} M_{|\be|}}{h_{m}(b_\rh' d_A(x))}.
\end{align*}

In analogy to (<ref>) we define the family $(\vh_j)_{j\ge 1}$ which clearly has the properties (1)–(3).
The fact that at most $12^{2n}$ factors in the product defining $\vh_j$ are different from $1$ together with the estimates for $\ps_j$
and (<ref>)
easily imply (4).

§ EXTENSION OF WHITNEY ULTRAJETS

If we restrict an $\cE^{[M]}$-function defined on $\R^n$ to a closed subset of $\R^n$, then Taylor's theorem
implies necessary conditions which lead to the notion of Whitney ultrajets.

§.§ Whitney ultrajets

Let $M=(M_k)$ be a weight sequence.
Let $K \subseteq \R^n$ be a compact set.
A Whitney ultrajet of class $\cE^{\{M\}}$ on $K$ is a Whitney jet
$F= (F^\al)_{\al} \in \cE(K)$ such that
there exist $C>0$ and $\rh \ge 1$

\begin{gather}
|F^\al(x)| \le C \rh^{|\al|}   M_{|\al|}, \quad \al \in \N^n,~ x \in K,
\label{jets1}
\\
|(R^p_x F)^\al(y)| \le C \rh^{p+1} \, |\al|!\, m_{p+1}\,  |x-y|^{p+1-|\al|}, \quad p \in \N,\, |\al| \le p,~ x,y \in K.
\label{jets2}
\end{gather}

A Whitney ultrajet of class $\cE^{(M)}$ on $K$ is a Whitney jet
$F= (F^\al)_{\al} \in \cE(K)$ with the property that for each $\rh>0$ there exists $C>0$ such that
(<ref>) and (<ref>) hold.

The set of Whitney ultrajets of class $\cE^{[M]}$ on $K$ is denoted by $\cE^{[M]}(K)$.
For a closed set $A \subseteq \R^n$ let

\[
\cE^{[M]}(A) := \{F \in \cE(A) : F|_K \in \cE^{[M]}(K) \text{ for all compact } K \subseteq A\}
\]

be the set of Whitney ultrajets of class $\cE^{[M]}$ on $A$.

We endow $\cE^{(M)}(A)$ with the projective limit topology with respect to the system of seminorms
$p_{K,1/\ell}(F) := \|F\|^M_{K,1/\ell} + |F|^M_{K,1/\ell}$,  $\ell \in \N_{\ge 1}$ and $K \subseteq A$ compact,

\begin{equation} \label{eq:jetseminorm1}
\|F\|^M_{K,\rh} := \sup_{\al \in \N^n} \frac{\|F^\al\|_K}{\rh^{|\al|} M_{|\al|}}
\end{equation}

and[Note that $|\al|!\, m_{p+1} \le \frac{M_{p+1}}{(p+1 -|\al|)!} \le 2^{p+1} |\al|!\, m_{p+1}$.]

\begin{equation} \label{eq:jetseminorm2}
| F |^M_{K,\rh}  := \sup_{\substack{x,y \in K\\ x \ne y}} \sup_{p \in \N } \sup_{|\al| \le p}  |(R^p_x F)^\al (y)|
\frac{(p+1-|\al|)!}{|x-y|^{p+1-|\al|} \rh^{p+1} M_{p+1}}.
\end{equation}

Similarly, the Roumieu space

\[
\cE^{\{M\}}(A) = \{F \in \cE(A) : \A K \subseteq_{cp} A \E m\in \N_{\ge 1} : p_{K,m}(F)<\infty \}
\]

is endowed with its natural locally convex topology.
Sometimes we shall need the Banach space $\cE^M_\rh(K) := \{F \in \cE(K): p_{K,\rh}(F)<\infty \}$.

Let $k$ be a positive integer.
A compact subset $K \subseteq \R^n$ is called Whitney $k$-regular if there is a constant $C >0$
such that for all $x,y \in K$ there exists a rectifiable path $\ga$ in $K$ joining $x$ and $y$ such that

\[
\on{length}(\ga) \le C |x-y|^{1/k}.
\]

It is not difficult to see that (<ref>) implies (<ref>) provided that $K$ is Whitney $1$-regular, e.g.,
if $K$ is convex.
This is no longer true if $K$ is only Whitney $k$-regular for some $k>1$; cf. <cit.>.

The goal of this section is to prove that Whitney ultrajets admit extensions preserving the class if the weight sequence $M$
is strongly regular. The proof of the Roumieu case is based on the existence of optimal cutoff functions and the extension problem for
the singleton (the Borel map). The Beurling case can be extracted from the Roumieu case by a reduction argument based on

In <Ref> we will prove that extension operators always exist in the Beurling case.
As a by-product it gives an alternative direct proof that extensions exist.

§.§ Extension of Whitney ultrajets of Roumieu type

Let $A \subseteq \R^n$ be a non-empty closed set and $F \in \cE(A)$ a Whitney jet on $A$.
A function $f \in C^\infty(\R^n)$ is said to be a local extension of $F$ in x if

\[
j^\infty_{\{x\}} f = F(x).
\]

The next lemma provides local extensions in a uniform way.

Let $M$ be a strongly regular weight sequence.
Let $K\subseteq \R^n$ be a compact non-empty set and $F \in \cE^{\{M\}}(K)$.
Then there exists $\rh>0$ such that
for all $x \in K$ there exists $f_x \in \cB^{\{M\}}(\R^n)$ with $j^\infty_{\{x\}} f_x = F(x)$ and

\begin{align} \label{eq:Wprep1}
\sup_{x \in K} \|f_x\|^M_{\R^n,\rh} < \infty.
\end{align}

By assumption, $\{F(x) : x \in K\}$ is a bounded subset of $\La^{\{M\}}$.
We know that $j^\infty_{\{0\}} : \cB^{\{M\}}(\R^n) \to \La^{\{M\}}$ is surjective, by <Ref>.
Since both spaces are Silva spaces, each bounded set in $\La^{\{M\}}$ is the image of a bounded set in $\cB^{\{M\}}(\R^n)$.
Thus, there is a bounded set $B$ in $\cB^{\{M\}}(\R^n)$ such that for all $x \in K$
there exists $f_x \in B$ with $j^\infty_{\{x\}} f_x = F(x)$.
This also implies (<ref>).

In the setting of the previous lemma,
there exist $D,\si>0$
such that for all
$x,y \in K$, $z \in \R^n$
and $\al \in \N^n$ we have

\begin{equation} \label{eq:prepW2}
|(f_x - f_y)^{(\al)}(z)| \le D \si^{|\al|} M_{|\al|}  h_m(\si(|z-x|+|z-y|)).
\end{equation}

Let $x,y \in K$ be fixed. Take $p\in \N$ and $|\al| \le p$.
We have

\begin{align*}
T_x^pF(z) - T_y^pF(z)
&=	\sum_{\be \le p} \frac{(z-x)^\be}{\be!} (T_x^pF - T_y^pF)^{(\be)}(x)
\\
&=	\sum_{\be \le p} \frac{(z-x)^\be}{\be!} (R_y^pF - R_x^pF)^{\be}(x)
= \sum_{\be \le p} \frac{(z-x)^\be}{\be!} (R^p_y F)^\be(x).
\end{align*}

Using (<ref>), we find that there exist $A,\ta>0$ such that

\begin{align*}
|(T_x^pF - T_y^pF)^{(\al)}(z)|
&\le A \ta^{p+1} M_{p+1} \frac{(|z-x|+|x-y|)^{p+1-|\al|}}{(p+1-|\al|)!}
\\
&\le A (2\ta)^{p+1} M_{p+1} \frac{(|z-x|+|z-y|)^{p+1-|\al|}}{(p+1-|\al|)!}.
\end{align*}

By Taylor's formula and $j^\infty_{\{x\}} f_x = F(x)$,

\begin{align*}
f_x(z) - T_x^p F(z) = \sum_{|\be| = p+1} \frac{(z-x)^\be}{\be!}  f_x^{(\be)}(x+t(z-x)), \quad  t \in [0,1],
\end{align*}

and thus

\begin{align} \label{eq:fxTxF}
|(f_x - T_x^pF)^{(\al)} (z)| \le C \rh^{p+1} M_{p+1} \frac{|z-x|^{p+1-|\al|}}{(p+1-|\al|)!}, \quad |\al| \le p.
\end{align}

It follows that, for suitable constants $D,\si_1>0$,

\begin{align*}
|(f_x - f_y)^{(\al)}(z)| \le D \si_1^{p+1} M_{p+1} \frac{(|z-x|+|z-y|)^{p+1-|\al|}}{(p+1-|\al|)!},
\end{align*}

for all $x,y \in K$, $z \in \R^n$, $p \in \N$, and
$|\al| \le p$. Using that $M$ has moderate growth, we infer

\begin{align*}
|(f_x - f_y)^{(\al)}(z)| \le D \si^{|\al|} M_{|\al|} \cdot (\si(|z-x|+|z-y|))^{p+1-|\al|} m_{p+1-|\al|}
\end{align*}

for all $p \ge |\al|$. This implies (<ref>).

Let $M$ be a strongly regular weight sequence.
For each non-empty compact set $K \subseteq \R^n$ the mapping $j^\infty_K : \cE^{\{M\}}(\R^n) \to \cE^{\{M\}}(K)$ is surjective.

Let $F \in \cE^{\{M\}}(K)$ satisfy (<ref>) and (<ref>).
By <Ref>, there are $f_x \in \cB^{\{M\}}(\R^n)$, $x \in K$, satisfying $j^\infty_{\{x\}}f_x = F(x)$ for all $x \in  K$
and (<ref>).
Let $(Q_j)_{j\ge 1}$ be a family of Whitney cubes for $K$.
Let $\ep>0$ (to be specified later).
By <Ref>, there is
a family of $C^\infty$-functions $(\vh_{j,\ep})_{j\ge 1}$ satisfying
$0 \le \vh_{j,\ep} \le 1$, $\on{supp} \vh_{j,\ep} \subseteq Q_j^*$, $\sum_j \vh_{j,\ep} =1$ on $\R^n \setminus K$,

\begin{equation} \label{estvh}
\le \frac{\ep^{|\be|} M_{|\be|}}{h_{m}(C_1\ep\, d_K(x))}, \quad \be \in \N^n,~ x \in \R^n\setminus K,
\end{equation}

where $C_1>0$ is independent of $\ep$.
Let $x_j$ denote the center of $Q_j$ and let
$\hat x_j \in K$ be such that $|x_j - \hat x_j|= d_K(x_j)$. We define

\begin{equation} \label{eq:extensionformula}
f(z) :=
\begin{cases}
\sum_{j \ge 1} \vh_{j,\ep}(z) \, f_{\hat x_j}(z),  & \text{ if } z \in \R^n \setminus K, \\
F^0(z), & \text{ if } z \in  K.
\end{cases}
\end{equation}

Clearly, $f$ is $C^\infty$ in $\R^n \setminus K$.

There exist constants $D_1,\si_1>0$ such that, for $x \in K$,

\begin{align} \label{eq:Wclaim}
|(f-f_x)^{(\al)} (z)| \le D_1 \si_1^{|\al|} M_{|\al|} h_m(\si_1 |z-x|), \quad z \in \R^n \setminus K.
\end{align}

We must estimate

\begin{align} \label{eq:toest}
(f-f_x)^{(\al)} (z) = \sum_{\be \le \al} \binom{\al}{\be} \sum_{j \ge 1} \vh^{(\be)}_{j,\ep}(z) \, (f_{\hat x_j} -f_x)^{(\al-\be)}(z).
\end{align}

Let us first treat the term with $\be =0$.
If $z \in \on{supp} \vh_{j,\ep} \subseteq Q_j^*$, then

\begin{align} \label{consequenceWcubes}
|z-\hat x_j|\le |z-x_j| + |x_j-\hat x_j|
\le \frac{9}{8} \on{diam} Q_j + d_K(x_j) \lesssim  d_K(z) \le   |z-x|,
\end{align}

by <Ref>(3)–(4).
Thus, by <Ref>, there exist $D,\si>0$ such that

\begin{align*}
\sum_{j\ge 1} \vh_{j,\ep}(z) \, \big|(f_{\hat x_j} -f_x)^{(\al)}(z)\big| \le D \si^{|\al|} M_{|\al|} h_m(\si |z-x|).
\end{align*}

Now we consider the terms with $\be \ne 0$ in (<ref>). To this end let $\hat z$ be a point in $K$ with $|\hat z - z| = d_K(z)$.
Since $\be \ne 0$, $\sum_{j \ge 1} \vh_{j,\ep}^{(\be)}(z) = 0$ and thus

\begin{align*}
\sum_{j\ge 1} \vh^{(\be)}_{j,\ep}(z) \, (f_{\hat x_j} -f_x)^{(\al-\be)}(z) = \sum_{j \ge 1} \vh^{(\be)}_{j,\ep}(z) \, (f_{\hat x_j} -f_{\hat z})^{(\al-\be)}(z).
\end{align*}

We infer from <Ref>, (<ref>), (<ref>), and <Ref>(5) that there exist constants $D,\si>0$ such that

\begin{align*}
\sum_{j\ge 1} |\vh^{(\be)}_{j,\ep}(z)| | (f_{\hat x_j} -f_{\hat z})^{(\al-\be)}(z)|
12^{2n} D\ep^{|\be|}\si^{|\al-\be|} M_{|\be|}  M_{|\al - \be|} \frac{h_m(\si d_K(z))}{h_{m}(C_1\ep\, d_K(z))}
\\
12^{2n} D\ep^{|\be|}\si^{|\al-\be|} M_{|\al|} h_m(C\si d_K(z))
\end{align*}

if we choose $\ep:= C\si/C_1$, where $C$ is the constant from (<ref>). The claim follows.

In order to show that $f$ is a $C^\infty$-function on $\R^n$ with $j^\infty_K f = F$, let, for $\al \in \N^n$,

\[
f^\al(z) :=
\begin{cases}
f^{(\al)}(z) & \text{ if } z \in \R^n \setminus K,
\\
F^\al(z) & \text{ if } z \in K.
\end{cases}
\]

We prove that $f = f^0$ is a $C^\infty$-function with $f^{(\al)} = f^\al$ for all $\al \in \N^n$.
First we show that all $f^\al$ are continuous.
This is clear near points $x \not\in K$. So let $x \in K$.
Now if $z \in K$, then

\[
|f^\al(z) - f^\al(x)| = |F^\al(z) - F^\al(x)| = |(R^{|\al|}_x F)^\al (z)| \to 0 \quad \text{ as } z\to x,
\]

by (<ref>).
If $z \not\in K$, then

\begin{align*}
|f^\al(z) - f^\al(x)| &= |f^{(\al)}(z) - f_x^{(\al)}(x)|
\\
&\le |f^{(\al)}(z) - f_x^{(\al)}(z)| + |f_x^{(\al)}(z) - f_x^{(\al)}(x)| \to 0  \quad \text{ as } z\to x,
\end{align*}

by (<ref>) and since $f_x^{(\al)}$ is continuous.
At this point we may refer to the lemma of Hestenes <cit.> or argue as follows:
If $\de_i \in \N^n$ denotes the multiindex with $(\de_i)_j = \de_{ij}$ (the Kronecker symbol), then for $x\in K$ and $z \in \R^n$,

\begin{align*}
|f^\al(z) - f^\al(x) - \sum_{i=1}^n (z_i - x_i) f^{\al+ \de_i}(x)| = |f^\al(z) - (T_x^{|\al|+1} F)^{(\al)}(z)| = o(|z-x|),
\end{align*}

by (<ref>), (<ref>), and (<ref>). It means that $f^\al$ is $C^1$ and $\p_{x_i} f^{\al} = f^{\al + \de_i}$.
The assertion follows.

Finally, we claim that $f \in \cE^{\{M\}}(\R^n)$. By (<ref>), it suffices to consider $z \not\in K$.
Then the claim follows from

\begin{align*}
|f^{(\al)}(z)| \le  |f_{\hat z}^{(\al)} (z)| +
|(f-f_{\hat z})^{(\al)} (z)|,
\end{align*}

(<ref>), and (<ref>).

§.§ Extension of Whitney ultrajets of Beurling type

The Beurling case may be reduced to the Roumieu case by means of the next lemma.

Let $M=(M_k)$ be a strongly regular weight sequence.
For any non-negative sequence $L=(L_k)$ with $L \lhd M$
there exists a strongly regular weight sequence $N =(N_k)$ with $L \preceq N \lhd M$.

There is a decreasing sequence $(\ep_k)_{k\ge 1}$ tending to $0$ with $L_k \le \ep_1\cdots \ep_k M_k$
for all $k \ge 1$; it suffices to take $\ep_k:= \sup_{j\ge k} (\frac{L_k}{M_k})^{1/k}$.
Since $M$ is strongly non-quasianalytic, there is a constant $C>0$ such that

\begin{equation} \label{eq:red1}
\sum_{j\ge k} \frac{1}{\mu_k} \le C \frac{k}{\mu_k}, \quad k \in \N.
\end{equation}

Applying <Ref> to $\al_k =\frac{1}{\mu_k}$, $\be_k = \max \{\ep_k,\frac{k}{\mu_k}\}$, and $\ga_k = \frac{k}{\mu_k}$,
we find a sequence $(\th_k)_{k \ge 1}$ with $\th_k \nearrow \infty$
such that

\begin{align} \label{eq:red2}
\sum_{j \ge k} \frac{\th_j}{\mu_j} \le 8 \th_k \sum_{j \ge k} \frac{1}{\mu_j}, \quad \text{ for all } k \ge 1,
\end{align}

$\th_k\ep_k \to 0$, and  $\frac{k\th_k}{\mu_k} \searrow 0$. Define
$\nu_k := \frac{\mu_k}{\th_k}$, $k \ge 1$, and $\nu_0 := 0$.
Then $N_k := \nu_0 \nu_1 \cdots \nu_k$ is a strongly log-convex weight sequence
that is strongly non-quasianalytic, by (<ref>) and (<ref>),
and has moderate growth,

\[
\frac{N_{j+k}}{N_jN_k} = \frac{M_{j+k}}{M_jM_k} \frac{\th_1\cdots\th_{j}  \th_1\cdots\th_{k}}{\th_1\cdots\th_{j+k}}
\le \frac{M_{j+k}}{M_jM_k}.
\]

To see $L \preceq N \lhd M$ observe that
$(\frac{L_k}{N_k})^{1/k} = (\frac{L_k}{M_k})^{1/k} (\th_1\cdots\th_{k})^{1/k} \le (\th_1\ep_1 \cdots\th_{k}\ep_k)^{1/k}$
is bounded and $(\frac{M_k}{N_k})^{1/k} = (\th_1 \cdots \th_k)^{1/k} \to \infty$.

Let $M$ be a strongly regular weight sequence.
For each non-empty compact set $K \subseteq \R^n$ the mapping $j^\infty_K : \cE^{(M)}(\R^n) \to \cE^{(M)}(K)$ is surjective.

Let $F = (F^\al)_\al \in \cE^{(M)}(K)$. Set $L_0:= \|F^0\|_K$ and

\[
L_k := \max \Big\{ \sup_{|\al|= k}\|F^\al\|_K, \sup_{x\ne y \in K,\, |\al|\le k-1}
\frac{k!\, |(R^{k-1}_x F)^\al(y)|}{|\al|! \, |y-x|^{k-|\al|}}  \Big\}, \quad k \ge 1.
\]

Then $L \lhd M$ and, by <Ref>, there is a strongly regular weight sequence
$N$ with $L \preceq N \lhd M$. So $F \in \cE^{\{N\}}(K)$ and it has an extension $f \in \cE^{\{N\}}(\R^n)$,
by <Ref>, which is also an element of $\cE^{(M)}(\R^n)$.

§.§ Extension from closed sets

It is now easy to get extension theorems from arbitrary closed subsets of $\R^n$.

Let $M$ be a strongly regular weight sequence.
For each non-empty closed set $A \subseteq \R^n$ the mapping $j^\infty_A : \cE^{[M]}(\R^n) \to \cE^{[M]}(A)$ is surjective.

This follows easily from <Ref>, since $\cE^{[M]}$ admits partitions of unity. Indeed,
fix $F \in \cE^{[M]}(A)$.
For $k \in \N_{\ge 1}$ set $U_k := \{x \in \R^n :  k-2 < |x| < k\}$; note that $U_k$, $k \ge 2$, are open shells and
$U_1 = \{x \in \R^n : |x|<1\}$.
There exist functions $\vh_k \in \cE^{[M]}(\R^n)$ such that $0 \le \vh_k \le 1$, $\on{supp} \vh_k \subseteq U_k$, and
$\sum_{k=1}^\infty \vh_k =1$.
For each $k \in \N_{\ge 1}$ the jet $F_k := F|_{\ol U_k}$ belongs to
$\cE^{[M]}(A\cap \ol U_k)$ and thus has an extension
$f_k \in \cE^{[M]}(\R^n)$, i.e.,
$j^\infty_{A\cap \ol U_k}(f_k) = F_k$.
$f := \sum_{k=1}^\infty \vh_k f_k$
is a function in $\cE^{[M]}(\R^n)$ since on any compact set the sum is finite.
Each $x \in A$ belongs to at most two consecutive sets $U_\ell$, $U_{\ell+1}$.
Thus, for each $\al \in \N^n$,

\begin{align*}
&= \sum_{\be \le \al} \binom{\al}{\be}\vh^{(\be)}_\ell(x) f^{(\al-\be)}_\ell(x)
+ \sum_{\be \le \al} \binom{\al}{\be}\vh^{(\be)}_{\ell+1}(x) f^{(\al-\be)}_{\ell+1}(x)
\\&= \sum_{\be \le \al} \binom{\al}{\be}\p^\be(\vh_\ell(x) + \vh_{\ell+1}(x))
=  F^\al(x),
\end{align*}

since all summands with $|\be|>0$ vanish.

§.§ Analytic extensions

The ultradifferentiable extension in <Ref> can be made real analytic
on the complement of $A$. This follows from a result of Schmets and Validivia [83].
A different proof due to Langenbruch <cit.> is based on a general
approximation theorem of Whitney type (cf. <cit.>) which is interesting in its own right.
In fact it is a special case of a quite general approximation theorem based
on a surjectivity criterion for continuous linear maps between Fréchet spaces.

Let $M=(M_k)$ be a non-quasianalytic weight sequence.
Let $\Om \subseteq \R^n$ be open and $\et: \Om \to (0,\infty)$ continuous.
For any $f \in \cE^{(M)}(\Om)$ there exists $g \in \cH(\Om^*)$ such that

\[
| f^{(\al)}(x) - g^{(\al)}(x) | \le \et(x)^{|\al|+1} M_{|\al|} \quad \text{ for all } x \in \Om, ~ \al \in \N^n.
\]

Here $\Om^* := \{ z \in \C^n : \Re (z) \in \Om, \, |\Im(z)| < \on{dist}(\Re (z) , \p \Om) \}$.
We may assume that $g|_\Om$ is real-valued.

Let $M=(M_k)$ be a non-quasianalytic weight sequence and $\Om \subseteq \R^n$ open.
There exists a continuous linear mapping $T : \cB^{[M]}(\R^n) \to \cB^{[M]}(\R^n)$ such that
$T(f)$ is an analytic $\Om$-modification of $f$, i.e.,
$T(f)|_\Om$ is real analytic and $j^\infty_A T(f) = j^\infty_A  f$, where $A:= \R^n \setminus \Om$.

§ CONTINUOUS LINEAR EXTENSION OPERATORS

In this section we address the question whether there exist continuous linear extension operators.
We concentrate on the Beurling case for which the question has an affirmative answer.
In <Ref> we comment briefly on the Roumieu case which has a negative answer most of the time.

Let $M=(M_k)$ be a strongly regular weight sequence.
Let $A \subseteq \R^n$ be a non-empty closed set.
By <Ref>, the following sequence is exact:

\begin{equation}\label{shortexactsequenceA}
\xymatrix{
0 \ar[r] & \ker j^\infty_A ~ \ar@{^{(}->}[r] & \cE^{(M)}(\R^n) \ar[r]^{j^\infty_A} & \cE^{(M)}(A) \ar[r] & 0
\end{equation}

We will show that the sequence (<ref>) splits: there exists
a continuous linear right-inverse $E_A : \cE^{(M)}(A) \to \cE^{(M)}(\R^n)$ of $j^\infty_A$,
that is $j^\infty_A \o E_A = \on{id}_{\cE^{(M)}(A)}$. We say that $E_A$ is an extension operator.

We give an elementary constructive proof based on the fact that the Borel map is split surjective
and on the existence of optimal cutoff functions.
In <Ref> we shall briefly comment on the alternative approach based on
the splitting theorem of Vogt and Wagner [91].

§.§ The existence of extension operators is a local property

We need a variant of <Ref>.

Let $M$ be a strongly regular weight sequence.
For each $\rh>0$ there exist $C,\si >0$ such that the following holds.
If $K \subseteq \R^n$ is compact and $H \subseteq \R^n$ is compact convex with $K \subseteq \on{int} H$,
$x_1,x_2 \in K$, and
$f_1,f_2 \in C^\infty(\R^n)$ and $F \in \cE(K)$ satisfy $j^\infty_{\{x_1\}} f_1 = F(x_1)$, $j^\infty_{\{x_2\}} f_2 = F(x_2)$,

\[
\|f_1\|^M_{H,\si} < \infty, \quad \|f_2\|^M_{H,\si} < \infty, \quad |F|^M_{K,\si} < \infty,
\]

then for all $z \in H \setminus K$ and $\al \in \N^n$,

\begin{equation*}
|(f_1 -f_2)^{(\al)}(z)| \le  (\|f_1\|^M_{H,\si} + \|f_2\|^M_{H,\si} + |F|^M_{K,\si}) \cdot C \rh^{|\al|} M_{|\al|} h_m(\rh (|z-x_1| + |z- x_2|)).
\end{equation*}

As in the proof of <Ref> we find, for $|\al| \le p$,

\[
|(T^p_{x_1}F - T^p_{x_2}F)^{(\al)}(z)| \le |F|^M_{K,\si} (2\si)^{p+1} M_{p+1} \frac{(|z-x_1| +|z-x_2|)^{p+1 -|\al|}}{(p+1 -|\al|)!}.
\]

Since $j^\infty_{\{x_i\}} f_i  = F(x_i)$,

\begin{align*}
|(f_i- T^p_{x_i} F)^{(\al)}(z)| \le \|f_i\|^M_{H,\si} \si^{p+1} M_{p+1} \frac{(|z-x_i|)^{p+1 -|\al|}}{(p+1 -|\al|)!}, \quad i=1,2,
\end{align*}

as in (<ref>).
Since $M$ has moderate growth, $M_{p+1} \le D^{p+1} M_{|\al|}M_{p+1 - |\al|}$ for some $D>0$. Then the triangle inequality gives
for all $p \ge |\al|$,

\begin{align*}
|(f_1 -f_2)^{(\al)}(z)| &\le (2\si D)^{|\al|} M_{|\al|} m_{p+1-|\al|} (2\si D (|z-x_1| + |z- x_2|))^{p+1 - |\al|}
\\
&\qquad \times	 (\|f_1\|^M_{H,\si} + \|f_2\|^M_{H,\si} + |F|^M_{K,\si})
\end{align*}

and the assertion follows easily.

Let $M$ be a strongly regular weight sequence.
Let $A \subseteq \R^n$ be a closed non-empty set.
A continuous linear mapping $E : \cE^{(M)}(A) \to \cE^{(M)}(\R^n)$ is said to be a local extension operator for $A$ in $x$

\begin{equation*}
j^\infty_{\{x\}} E(F) =  F(x), \quad \text{ for all } F \in \cE^{(M)}(A).
\end{equation*}

Clearly, $E$ is an extension operator for $A$ if it is a local extension operator for $A$ in every $x \in A$.

Let $M$ be a strongly regular weight sequence.
Let $A \subseteq \R^n$ be a closed non-empty set.
Suppose that for each $x \in \p A$ there exists a local extension operator $E_x$ for $A$ in $x$
and that $\{E_x : x \in \p A\}$ is locally equicontinuous in $L(\cE^{(M)}(A), \cE^{(M)}(\R^n))$.
Then there exists an extension operator $E : \cE^{(M)}(A) \to \cE^{(M)}(\R^n)$.

Let $(Q_j)_{j\in \N}$ be a family of Whitney cubes for $A$ (cf. <Ref>) and let $(\vh_j)_{j\in \N}$ be the
partition of unity provided by <Ref>.
For each $j$ choose a point $x_j \in \p A$ such that $d_{Q_j}(x_j)=\on{dist}(A,Q_j)$.
For $F \in \cE^{(M)}(A)$ define

\[
E(F)(z) :=
\begin{cases}
\sum_{j\ge1} \vh_j(z) E_{x_j}(F)(z) & \text{ if } z \in \R^n \setminus A,
\\
F^0(z) & \text{ if } z \in A.
\end{cases}
\]

Clearly, $E(F)$ is of class $C^\infty$ on $\R^n \setminus A$.
Recall that $p_{K,\si}(F) := \|F\|^M_{K,\si} + |F|^M_{K,\si}<\infty$ for all compact $K \subseteq A$ and $\si>0$.

For each compact convex $H \subseteq \R^n$
and for each $\rh>0$
there exist $C,\si,\ta>0$ and a compact subset $K \subseteq A$
such that
for all $x \in \p A \cap H$, $z \in H \setminus A$ with $|x-z|\le \frac{b_0}{7}$ (with $b_0$ from <Ref>),
$F \in \cE^{(M)}(A)$, and $\al \in \N^n$,

\begin{equation} \label{eq:estclaim}
|(E(F) - E_x(F))^{(\al)}(z)| \le C\, p_{K,\si}(F)\, \rh^{|\al|} M_{|\al|} h_m(\ta|x-z|).
\end{equation}

Let us prove the claim.
By <Ref>, there is $C>1$ such that

\begin{equation} \label{hmgR}
h_m(t) \le h_m(Ct)^2, \quad t>0.
\end{equation}

Let a compact convex set $H \subseteq \R^n$ and $\rh>0$ be fixed.
By <Ref>, there exist positive $C_1 =C_1(\rh)$ and $\ta_1 = \ta_1(\rh)$ such that, for all $j$,

\begin{equation} \label{eq:estcutoffB}
|\vh_j^{(\al)}(z)| \le C_1 \frac{\rh^{|\al|} M_{|\al|}}{h_m(\ta_1 d_A(z))},\quad \al\in \N^n,\, z \in \R^n.
\end{equation}

\begin{equation} \label{eq:rhotwo}
\rh_2 := \min\{\rh, \tfrac{b_0\ta_1}{7C} \}.
\end{equation}

We invoke <Ref> for $\rh_2$: there exist
$C_2,\si_2>0$ such that the assertion of the lemma
holds for $\rh_2$, $C_2$, $\si_2$ (instead of $\rh$, $C$, $\si$ in the lemma).

For $\ep>0$ let $H_\ep := \{z \in \R^n : d_H(z) \le \ep\}$ be the closed $\ep$-neighborhood of $H$.
By assumption, the set $\{E_x : x \in \p A \cap H_2\}$ is equicontinuous in $L(\cE^{(M)}(A), \cE^{(M)}(\R^n))$.
It follows that there exist constants $\si_3 \ge \si_2$, $C_3\ge 1$ and a compact set $K \subseteq A$ with $A \cap H_2 \subseteq K$
such that

\begin{equation*}
\|E_x(F)\|^M_{H_2,\si_2} \le C_3 \, p_{K,\si_3}(F), \quad F \in \cE^{(M)}(A), ~ x \in \p A \cap H_2.
\end{equation*}

Then, by the assertion of <Ref>, for all $x,y \in \p A \cap H_1$, $z \in H_1 \setminus A$ and $\al \in \N^n$,

\begin{align} \label{eq:equic}
\MoveEqLeft
|(E_x(F) - E_y(F))^{(\al)}(z)| \le C_4\, p_{K,\si_3}(F)\, \rh_2^{|\al|} M_{|\al|}  h_m(\rh_2(|z-x|+|z-y|)),
\end{align}

for some $C_4 \ge 1$.

We are ready to estimate

\[
(E(F) - E_x(F))^{(\al)}(z) = \sum_{\be \le \al} \binom{\al}{\be} \sum_{j \ge 1} \vh_j^{(\be)}(z) (E_{x_j}(F) - E_x(F))^{(\al-\be)}(z),
\]

where $x \in \p A \cap H$ and $z \in H \setminus A$ with $|x-z|\le \frac{b_0}{7}$.
By <Ref>, for each $j \ge 1$ and $z \in Q_j^*$,

\begin{align*}
|x_j - z| &\le d_{Q_j}(x_j) + \on{diam} Q_j^*
= \on{dist}(A,Q_j) + \tfrac{9}{8} \on{diam} Q_j \le 6 \on{diam} Q_j \le \tfrac{6}{b_0} d_A(z).
\end{align*}

Consequently (since $b_0 \le 1$),

\[
|x-x_j| \le |x-z| + |z-x_j| \le |x-z| + \tfrac{6}{b_0} d_A(z)  \le \tfrac{7}{b_0} |x-z| \le 1
\]

and thus $x_j \in H_1$.
So, by (<ref>), for the term with $\be =0$ we have

\begin{align*}
\MoveEqLeft
\sum_{j \ge 1} |\vh_j(z)| |(E_{x_j}(F) - E_x(F))^{(\al)}(z)|
\\
C_4 \, p_{K,\si_3}(F)\, \rh_2^{|\al|} M_{|\al|}  h_m(\rh_2(|z-x|+|z-x_j|))
\\
C_4 \, p_{K,\si_3}(F)\, \rh_2^{|\al|} M_{|\al|}  h_m(\tfrac{7\rh_2}{b_0} |z-x|)
\\
C_4 \, p_{K,\si_3}(F)\, \rh^{|\al|} M_{|\al|}  h_m(\ta_1 |z-x|).
\end{align*}

To estimate the terms with $\be \ne 0$, let $y \in \p A$ be such that $|y-z| = d_A(z)$.
Then $\sum_{j \ge 1} \vh_j^{(\be)}(z) = 0$ implies

\begin{align*}
\sum_{j\ge 1} \vh^{(\be)}_{j}(z) \, (E_{x_j}(F) -E_x(F))^{(\al-\be)}(z) = \sum_{j\ge 1} \vh^{(\be)}_{j}(z) \, (E_{x_j}(F) -E_y(F))^{(\al-\be)}(z).
\end{align*}

We have $|y -z| = d_A(z) \le |z-x| \le \frac{b_0}7$ and thus $y \in H_1$.
Moreover, if $z \in Q_j^*$, then

\[
|z-x_j| + |z-y| \le \tfrac{6}{b_0} d_A(z) + d_A(z)  \le \tfrac{7}{b_0} d_A(z) \le \tfrac{7}{b_0} |x-z| \le 1.
\]

Thus, by (<ref>) and (<ref>),

\begin{align*}
\MoveEqLeft
\sum_{j \ge 1} |\vh_j^{(\be)}(z)| |(E_{x_j}(F) - E_y(F))^{(\al-\be)}(z)|
\\
&\le 12^{2n} C_1 C_4 \, p_{K,\si_3}(F)\, \rh^{|\be|} M_{|\be|} \rh_2^{|\al-\be|} M_{|\al-\be|} \frac{ h_m(\rh_2(|z-x_j|+|z-y|))}{h_m(\ta_1 d_A(z))}
\\
&\stackrel{\eqref{eq:rhotwo}}{\le} 12^{2n} C_1 C_4 \, p_{K,\si_3}(F)\, \rh^{|\al|} M_{|\al|}  \frac{ h_m(\frac{\ta_1}{C}d_A(z))}{h_m(\ta_1 d_A(z))}
\\
&\stackrel{\eqref{hmgR}}{\le} 12^{2n} C_1 C_4 \, p_{K,\si_3}(F)\, \rh^{|\al|} M_{|\al|}  h_m(\ta_1 d_A(z)).
\end{align*}

Then the claim follows easily.

We are ready to complete the proof of the proposition.
That $E(F)$ is a $C^\infty$-function on $\R^n$ with $j^\infty_A E(F) = F$ follows in analogy to the proof of <Ref>.

It remains to show that $E(F) \in \cE^{(M)}(\R^n)$ and that the linear map $E : \cE^{(M)}(A) \to \cE^{(M)}(\R^n)$ is continuous.
Let $H \subseteq \R^n$ be a compact convex set and $\rh>0$.
Let $H_{b_0/7}$ be the closed $\tfrac{b_0}7$-neighborhood of $H$.
By the claim, there exist $C,\si,\ta>0$ and a compact set $K \subseteq A$ such that for all $x \in \p A \cap H_{b_0/7}$, $z \in H_{b_0/7} \setminus A$
with $|x-z| \le \tfrac{b_0}7$, $F \in \cE^{(M)}(A)$ and $\al \in \N^n$ we have (<ref>), in particular,

\begin{equation} \label{eq:estclaimR}
|(E(F) - E_x(F))^{(\al)}(z)| \le C\, p_{K,\si}(F)\, \rh^{|\al|} M_{|\al|}.
\end{equation}

If $z \in H \setminus A$ satisfies $d_A(z) \le \tfrac{b_0}7$, then there is $x \in \p A \cap H_{b_0/7}$ with $d_A(z) = |x-z|$.
For such $z$ and $\al \in \N^n$,

\begin{align*}
|E(F)^{(\al)}(z)| &\le |E_x(F)^{(\al)}(z)| + |E(F)^{(\al)}(z) - E_x(F)^{(\al)}(z)|
\\
&\le C \, p_{K,\si}(F)\,  \rh^{|\al|} M_{|\al|},
\end{align*}

by (<ref>) and by the assumption on local equicontinuity of $E_x$ (increasing $K$, $C$, and $\si$ if necessary).
Together with (<ref>) this shows the assertion.
The proof is complete.

§.§ Extension operators always exist in the Beurling case

Let $M$ be a strongly regular weight sequence.
Let $A \subseteq \R^n$ be a non-empty closed set.
There exists an extension operator $E : \cE^{(M)}(A) \to \cE^{(M)}(\R^n)$.

This follows from <Ref> and <Ref>.

§.§ A different approach based on a splitting theorem for Fréchet spaces

Let $M=(M_k)$ be a strongly regular sequence.
Let $K$ and $L$ be a non-empty compact subsets of $\R^n$ such that $K \subseteq \on{int} L$.
By <Ref>, the sequence

\begin{equation}\label{shortexactsequence}
\xymatrix{
0 \ar[r] & \cD^{(M)}(L,K)~ \ar@{^{(}->}[r] & \cD^{(M)}(L) \ar[r]^{j^\infty_K} & \cE^{(M)}(K) \ar[r] & 0
\end{equation}

is an exact sequence of Fréchet spaces,
where we endow

\[
\cD^{(M)}(L,K) := \{f \in \cE^{(M)}(\R^n) : \on{supp} f \subseteq L,\, j^\infty_K f = 0\}
\]

and $\cD^{(M)}(L)$
with the subspace topology of $\cE^{(M)}(\R^n)$.
By the splitting theorem of Vogt and Wagner [91] (see also <cit.>),
the sequence splits if $\cE^{(M)}(K)$ has property (DN) and
has property ($\Om$).

A Fréchet space $E$ with fundamental system of seminorms $\sP = (p_j)_{j \in \N}$
has property (DN) if

\[
\E m \in \N \A k \in \N \E \ell \in \N \E C>0 : p_k^2 \le C p_m p_\ell.
\]

Then $p_m$ is a norm and every norm with this property is called a dominating norm.
It has property ($\Om$) if

\[
\A m \in \N \E k \in \N \A \ell \in \N \E C>0 \E \th \in (0,1) : p_k^* \le C  (p_m^*)^{1-\th} (p_\ell^*)^\th,
\]

where $p^*_k(y) := \sup\{|y(x)| : p_k(x)\le 1\}$ is the dual norm of $p_k$.
The properties (DN) and ($\Om$) are linear topological invariants.
The property (DN) is inherited by all closed subspaces.[Hence, the splitting of
(<ref>) implies that $\cE^{(M)}(K)$ has the property (DN).]
A nuclear Fréchet space $E$ has property (DN) if and only if $E$ is isomorphic to a closed subspace of $s$ (the space of rapidly decreasing sequences).
The property ($\Om$) is inherited by all quotient spaces.
A nuclear Fréchet space $E$ has property ($\Om$) if and only if $E$ is isomorphic to a quotient space of $s$.
Cf. <cit.>.

That $\cE^{(M)}(K)$ has property (DN) is a triviality.
The fundamental system of seminorms

\[
p_m(F):= \|F\|^M_{K,1/m} + |F|^M_{K,1/m}, \quad m \in \N_{\ge 1},
\]

cf. (<ref>) and (<ref>), satisfies $p_k^2 \le p_1 p_{k^2}$ for all $k\ge 1$,

\begin{align*}
\Big(\frac{k^{|\al|}}{M_{|\al|}} \Big)^2 = \frac{k^{2|\al|}}{M_{|\al|}} \frac{1^{|\al|}}{M_{|\al|}}.
\end{align*}

That $\cD^{(M)}(L,K)$ has property ($\Om$) follows from a result by Franken [37] (building on work of Meise and Taylor [63]).
Both papers are situated in the setting of Braun–Meise–Taylor classes of which the Denjoy–Carleman classes form a subclass if $M=(M_k)$ is a
strongly regular weight sequence; see <Ref>(2).

§.§ Continuous linear extension operators in the Roumieu case

As already pointed out in <Ref> in the case of the singleton, extension operators rarely exist in the Roumieu case.
For instance, the condition (<ref>) due to [68]
shows that $j^\infty_{\{0\}} : \cE^{\{M\}}(\R^n) \to \cE^{\{M\}}(\{0\}) = \La^{\{M\}}$ has no continuous linear right-inverse
if $M$ is any Gevrey sequence $M_k = G^s_k = k!^s$, $s>1$.

Actually, $j^\infty_{\{0\}} : \cE^{\{M\}}([-1,1]^n) \to \La^{\{M\}}$ is not split surjective
for any strongly regular weight sequence $M$ as seen by the following topological argument (cf. <cit.>):
$\cE^{\{M\}}([-1,1]^n)$ is isomorphic to the dual of a power series space of finite type (cf. [52]),
which cannot have $\La^{\{M\}}$ as a complemented subspace, since $\La^{\{M\}}$ is isomorphic to the
dual of a power series space of infinite type (cf. <cit.>).

For the Whitney problem, Langenbruch [51] proved that
for compact convex subsets $K,L$ of $\R^n$ with $\emptyset \ne \on{int} K \subseteq K \subseteq \on{int} L$
there is no extension operator $E : \cE^{\{M\}}(K) \to \cE^{\{M\}}(L)$ if
$M$ is a weight sequence satisfying

\begin{gather}
\E a >0 \A b \in \N_{\ge a} :
\ol \la(b):= \limsup_{k \to \infty} \frac{\mu_{bk}}{\mu_k} \le a  \liminf_{k \to \infty} \frac{\mu_{bk}}{\mu_k}
=: a\, \ul \la(b), \label{eq:Lreg1}
\\
\lim_{b \to \infty} \frac{\ul \la(b)}{b} = \infty. \label{eq:Lreg2}
\end{gather}

A weight sequence with these properties is strongly regular (cf. <cit.>
and <cit.>). It is easy to check that the Gevrey sequences $G^s$, $s>1$, have the properties
(<ref>) and (<ref>) (indeed, in that case $\ol \la(b) = \ul \la(b) = b^s$).

§.§ A different approach based on polynomial approximation

Denote by $\sP_d$ the space of all polynomials on $\R^n$ of degree at most $d$, and let $\sP_{-1} := \{0\}$.
For compact $K \subseteq \R^n$ and
continuous $f : K \to \R$ we set

\[
\on{dist}_K(f,\sP_d) := \inf\{\|f - p\|_K : p \in \sP_d\}.
\]

Let $M=(M_k)$ be a weight sequence.
Let $\cA^{\{M\}}(K)$ be the set of all $f \in C^0(K)$ such that there exists $\rh>0$ such that

\begin{equation} \label{eq:seminormapprox}
\|f\|^{\cA^M}_{K,\rh}:= \sup_{k \in \N} \sup_{d \ge -1} \frac{(d+2)^k  \on{dist}_K(f,\sP_d)}{\rh^k M_k} < \infty
\end{equation}

and $\cA^{(M)}(K)$ the set of all $f \in C^0(K)$ such that (<ref>) holds for all $\rh>0$.
The spaces $\cA^{[M]}(K)$ are endowed with their natural locally convex topologies.

The following theorem is an ultradifferentiable version of Jackson's theorem.

Let $M$ be a derivation-closed weight sequence with $m_k^{1/k} \to \infty$.
For any compact $K \subseteq \R^n$ we have $j^0_K \cE^{[M]}(\R^n) \subseteq \cA^{[M]}(K)$, where $j^0_K(f) = f|_K$.
The inclusion is continuous.

Let $r \ge 1$.
A compact set $K \subseteq \R^n$ is said to have the Markov property $(P_r)$ if
there exist $C_1,C_2>0$ such that
for every polynomial $p$[It is not hard to see that $(P_r)$ holds if and only if there is $C>0$ such that
$\|p^{(\al)}\|_K \le C \deg(p)^{r|\al|} \|p\|_K$ for all $|\al|\ge 1$ and all polynomials $p$;
cf. <cit.>.
Any compact subset of $\R^n$ that is uniformly polynomially cuspidal has the Markov property,
in particular, any fat subanalytic compact subset of $\R^n$.

\begin{equation} \label{eq:Markov}
|p(z)| \le C_1 \|p\|_K \quad \text{ for } z \in \C^n \text{ with } \on{dist}(z,K) \le \frac{C_2}{\deg(p)^r}.
\end{equation}

If $K\subseteq \R^n$ satisfies $(P_r)$, then necessarily $r\ge 2$, by [5],
and $K$ is $C^\infty$-determining, see <cit.>,
that means $j^0_K f =0$ implies $j^\infty_K f =0$ for any $C^\infty$-function $f$.

The following extension theorem involves an unavoidable loss of regularity: for instance
[39] shows that $j^0_{[-1,1]} : \cE^{(G^s)}(\R) \to \cA^{(G^s)}([-1,1])$ is not surjective.
On the other hand the weight sequence is not required to be strongly non-quasianalytic.

Let $M$ be a non-quasianalytic weight sequence of moderate growth.
Let $K \subseteq \R^n$ be a compact set satisfying $(P_r)$ with some $r\ge 2$.
Then there exists a continuous linear operator
\(L : \cA^{[M]}(K) \to \cE^{[M^{r+1}]}(\R^n)\)
such that $j^0_K \o L = \on{id}$. Here $M^{r+1} := (M^{r+1}_k)$ is the $(r+1)$-st component-wise power of $M$.

By <Ref>, there is a sequence $(\vh_k)_{k\ge 1}$ of functions $\vh_k \in C^\infty(\R^n)$
such that $0 \le \vh_k \le 1$, $\on{supp} \vh_k \subseteq U_k := \{x \in \R^n : d_K(x)< C_2 k^{-r}\}$
(where $C_2$ is the constant from (<ref>)),
$\vh_k=1$ in a neighborhood of $K$, and

\[
|\vh_k^{(\al)}| \le C(n)^{|\al|} k^{r|\al|} M_{|\al|}, \quad \al \in \N^n.
\]

For $f \in \cA^{[M]}(K)$ let $L_d f$ be the Lagrange interpolation polynomial of $f$ with
nodes in Fekete–Leja extremal points of $K$ of order $d$; cf. <cit.>.

\[
\|f- L_df\|_K \le 2 (d+1)^n \on{dist}_K(f,\sP_d).
\]

\[
Lf(x) := \vh_1(x)L_1f(x) + \sum_{d=1}^\infty \vh_d(x) (L_{d+1}f(x)- L_df(x))
\]

and let us check that $L$ is well-defined and has the desired properties.
By (<ref>),

\begin{align*}
|\p^\al &(\vh_d(x) (L_{d+1}f(x)- L_df(x)))|
\\
\sum_{\be \le \al} \binom{\al}{\be} |\vh_d^{(\be)}(x)| |(L_{d+1}f- L_df)^{(\al-\be)}(x)|
\\
C_1 \sum_{\be \le \al} \binom{\al}{\be} \|\vh_d^{(\be)}\|_{U_d} \|(L_{d+1}f- L_df)^{(\al-\be)}\|_K
\\
C C_1 \sum_{\be \le \al} \binom{\al}{\be} C(n)^{|\be|} d^{r|\be|} M_{|\be|} \cdot  (d+2)^{n+r|\al-\be|} \on{dist}_K(f,\sP_d)
\\
&\le D \si^{|\al|} (d+2)^{n+r|\al|} M_{|\al|} \on{dist}_K(f,\sP_d),
\end{align*}

for suitable constants $D,\si>0$.
Thus, by (<ref>) and moderate growth of $M$,

\begin{align*}
\|\p^\al (\vh_d (L_{d+1}f- L_df))\|_{\R^n}
&\le D \si^{|\al|} (d+2)^{n+r|\al|} M_{|\al|}\cdot  \|f\|^{\cA^M}_{K,\rh}  \frac{\rh^{n+2+r|\al|} M_{n+2+r|\al|}}{(d+2)^{n+2+r|\al|}}
\\
&\le D_1 \si_1^{|\al|}  M^{r+1}_{|\al|} \|f\|^{\cA^M}_{K,\rh} \cdot \frac{1}{(d+2)^{2}}.
\end{align*}

It follows that $Lf$ is a well-defined $C^\infty$-function satisfying

\[
\|(Lf)^{(\al)}\|^{M^{r+1}}_{\R^n, \si_1} \le D_2 \, \|f\|^{\cA^M}_{K,\rh},
\]

which implies the assertion; notice for the Beurling case that $\si_1 \to 0$ as $\rh \to 0$.

§ EXTENSION WITH CONTROLLED LOSS OF REGULARITY

If the weight sequence $M=(M_k)$ is not strongly non-quasianalytic, and thus extension preserving the class is not possible,
it is natural to ask if the loss of regularity occurring in the extension can at least be controlled.
This is indeed the case.
It requires a new extension technique which is inspired by Dyn'kin's theory of almost analytic functions;
cf. [33, 34] and also [40].
For the singleton see <Ref>.

§.§ The setup: admissible pairs of weight sequences

Let $M,N$ be strongly log-convex weight sequences, $M$ of moderate growth with $m_k^{1/k}\to \infty$, $N$ non-quasianalytic with
$M \preceq N$, and