# On generalized definitions of ultradifferentiable classes

Javier Jiménez-Garrido, David Nicolas Nenning and Gerhard Schindl  J. Jiménez-
Garrido: Departamento de Matemáticas, Estadística y Computación, Universidad
de Cantabria, Avda. de los Castros, s/n, 39005 Santander, Spain. Instituto de
Investigación en Matemáticas IMUVA, Universidad de Valladolid, Spain.
<EMAIL_ADDRESS>D. N. Nenning: Fakultät für Mathematik,
Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria.
<EMAIL_ADDRESS>G. Schindl: Fakultät für Mathematik,
Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria.
<EMAIL_ADDRESS>
###### Abstract.

We show that the ultradifferentiable-like classes of smooth functions
introduced and studied by S. Pilipović, N. Teofanov and F. Tomić are special
cases of the general framework of spaces of ultradifferentiable functions
defined in terms of weight matrices in the sense of A. Rainer and the third
author. We study classes “beyond geometric growth factors” defined in terms of
a weight sequence and an exponent sequence, prove that these new types admit a
weight matrix representation and transfer known results from the matrix-type
to such a non-standard ultradifferentiable setting.

###### Key words and phrases:

Classes of ultradifferentiable functions, weight sequences and weight matrices

###### 2020 Mathematics Subject Classification:

26A12, 26A48, 46A13, 46E10

J. Jiménez-Garrido is supported by project PID2019-105621GB-I00, D.N. Nenning
and G. Schindl are supported by FWF-Project P33417-N

##  1\. Introduction

Spaces of ultradifferentiable functions are sub-classes of smooth functions
with certain restrictions on the growth of their derivatives. Two classical
approaches are commonly considered, either the restrictions are expressed by
means of a weight sequence $M=(M_{j})_{j}$, also called Denjoy-Carleman
classes (e.g. see [11]), or by means of a weight function $\omega$ also called
Braun-Meise-Taylor classes; see [3]. More precisely (in the one-dimensional
case) for each compact set $K$, the sets

(1.1) $\left\\{\frac{f^{(j)}(x)}{h^{j}M_{j}}\,\,:\,\,j\in\mathbb{N},\,\,x\in
K\right\\},\quad\text{respectively}\quad\left\\{\frac{f^{(j)}(x)}{\exp(\frac{1}{h}\varphi^{*}_{\omega}(hj))}\,\,:\,\,j\in\mathbb{N},\,\,x\in
K\right\\},$

are required to be bounded, where $\varphi^{*}_{\omega}$ denotes the Young-
conjugate of $t\mapsto\omega(e^{t})$. We shall mention that in the second
situation the classes can be defined directly by using $\omega$ and
controlling the decay of the Fourier transform $\widehat{f}$ with growth
factors $t\mapsto\exp(h\omega(t))$, $h>0$. In fact, this is the original
description; see [1] and also the discussion in [3] where the original
approach is transferred to the boundedness condition expressed in (1.1).

In the literature standard growth and regularity conditions are assumed for
$M$ and $\omega$ and in both settings we can consider two different types of
spaces: For the Roumieu-type the boundedness of the sets in (1.1) is required
for some $h>0$, whereas for the Beurling-type it is required for all $h>0$.

The most well-known examples are the Gevrey sequences of type $\alpha>0$ with
$G^{\alpha}_{j}:=j^{\alpha j}$ for $j\in\mathbb{N}$ (or equivalently use
$M^{\alpha}_{j}:=j!^{\alpha}$). Alternatively, one can use the function
$t\mapsto t^{1/\alpha}=:\omega_{\alpha}(t)$.

It is then a natural question how both classical settings are related. In [2]
this problem is studied and it has been shown that in general both approaches
are mutually distinct. However, based on this work, in [27] and [23] A. Rainer
and the third author have introduced the notion of weight matrices
$\mathcal{M}=\\{M^{(x)}:x>0\\}$ which allows to treat both classical methods
in a unified way and to transfer proofs from one context to the other. This
can be achieved when considering $\mathcal{M}=\\{M\\}$ for the weight sequence
and the so-called associated weight matrix
$\mathcal{W}:=\\{W^{(\ell)}:\ell>0\\}$ with
$W^{(\ell)}_{j}:=\exp(\frac{1}{\ell}\varphi^{*}_{\omega}(\ell j))$ in the
weight function case. But one is also able to describe more classes, e.g. take
the Gevrey matrix $\mathcal{G}:=\\{G^{\alpha}:\alpha>1\\}$; see [23, Thm.
5.22].

A second recent generalization was presented by S. Pilipović, N. Teofanov and
F. Tomić; see [19]. For given parameters $\tau>0$ and $\sigma>1$ they consider
the sequence $M^{\tau,\sigma}_{j}:=j^{\tau j^{\sigma}}$. However, in their
definition the geometric growth factor $h^{j}$ appearing in (1.1) is replaced
by $h^{j^{\sigma}}$. Observe that the growth of $j\mapsto h^{j^{\sigma}}$ is
closely connected with $j\mapsto M^{\tau,\sigma}_{j}$. The authors called
their framework “beyond Gevrey regularity” because $M^{\tau,1}_{j}=j^{\tau j}$
for $\sigma=1$, i.e., the Gevrey sequence of type $\tau>0$. Since all the
classes considered in this work are, in some sense, generalizations of Gevrey
classes, these spaces will be called Pilipović-Teofanov-Tomić classes, or PTT-
classes for short.

The difference between the growth of $j\mapsto h^{j}$ and $j\mapsto
h^{j^{\sigma}}$ suggests that the PTT-classes can be viewed as “non-standard
ultradifferentiable classes” and one can ask how both generalizations are
related. In the introduction of [21] it was claimed that the PTT-classes are
not covered by the weight matrix approach which is due to the different growth
of the factors mentioned before. However, the aim of this paper is to show
that also the PTT-classes are contained in the weight matrix approach.

In fact, we treat a more abstract setting by considering an exponent sequence
$\Phi=(\Phi_{j})_{j\in\mathbb{N}}$ and by replacing in (1.1) the growth
$j\mapsto h^{j}$ by $j\mapsto h^{\Phi_{j}}$. This notion yields
“ultradifferentiable classes beyond geometric growth factors” and we show that
under mild regularity and growth assumptions on $\Phi$ such spaces admit a
representation as weight matrix classes (as locally convex vector spaces) by
involving the canonical matrix

$\mathcal{M}_{M,\Phi}:=\\{M^{(c,\Phi)}:c>0\\},\quad
M^{(c,\Phi)}_{j}:=c^{\Phi_{j}}M_{j}.$

Applying this main result to the PTT-classes, we are also able to see that
when both $\sigma>1$ and $\tau>0$ are fixed then the corresponding space
cannot be represented by a single weight sequence $M$ or by a weight function
$\omega$; i.e., one requires the general weight matrix setting to describe
these classes. In other words PTT-classes constitute genuine examples of
ultradifferentiable classes defined by weight matrices.

On the other hand, in the very recent paper [33] it is shown that when only
$\sigma>1$ is fixed and when one considers matrix-type classes with parameter
$\tau>0$, i.e. PTT-limit classes, then these spaces can alternatively be
defined in terms of a weight function (in particular of a so-called associated
weight function). We give an independent proof of this result by applying
purely weight matrix techniques; see Theorem 6.8.

The paper is structured as follows: In Section 2 all necessary and relevant
conditions on weight sequences, weight functions and weight matrices are given
and the corresponding classes are defined. In Section 3 we introduce
ultradifferentiable spaces “beyond geometric growth factors” and prove in
Section 4 the main characterization results, i.e., Theorems 4.4, 4.7 and 4.9,
showing that, in particular, the PTT-classes can be represented as weight
matrix spaces. In Section 5 we apply this fact for fixed parameters $\tau>0$,
$\sigma>1$, and study properties of the relevant matrix
$\mathcal{M}^{\tau,\sigma}$ in order to transfer known results from the matrix
setting to PTT-classes. In Section 6 this is done analogously for so-called
limit classes when fixing $\sigma$ but letting $\tau\rightarrow 0$ resp.
$\tau\rightarrow+\infty$. It is shown that such spaces can be represented as
Braun-Meise-Taylor classes (see Theorem 6.8) and satisfy additional properties
since in this weight structure both mixed moderate growth conditions of the
particular type are valid.

### Acknowledgements

We wish to thank J. Vindas for pointing out additional results available for
matrix classes; more precisely for bringing, what is now property (f) in
Sections 5.2 and 6.3, to our attention. In addition he suggested to consider
[4], whose implications are the content of Section 6.4.

And we thank N. Teofanov and F. Tomić for forwarding their preprint of [33]
and the subsequent helpful discussions.

##  2\. Weights and conditions

###  2.1. General notation

We write $\mathbb{N}:=\\{0,1,2,\dots\\}$ and
$\mathbb{N}_{>0}:=\\{1,2,3,\dots\\}$. With $\mathcal{E}$ we denote the class
of all smooth functions. We use the standard multi-index notation and
$f^{(\alpha)}$, $\alpha\in\mathbb{N}^{d}$, stands for the $\alpha$-th
derivative of a given smooth function $f$ (defined in $\mathbb{R}^{d}$).

Occasionally, we write the symbol $[\cdot]$ if we mean either $\\{\cdot\\}$
(Roumieu-type) or $(\cdot)$ (Beurling-type) for spaces and growth conditions.

###  2.2. Weight sequences

Given a sequence $M=(M_{j})_{j}\in\mathbb{R}_{>0}^{\mathbb{N}}$ we also use
$m=(m_{j})_{j}$ defined by $m_{j}:=\frac{M_{j}}{j!}$ and
$\mu_{j}:=\frac{M_{j}}{M_{j-1}}$, $j\geq 1$, and set $\mu_{0}:=1$. Analogously
these conventions are used for all other appearing sequences, i.e.,
$N\leftrightarrow n\leftrightarrow\nu$ etc. $M$ is called normalized if
$1=M_{0}\leq M_{1}$ holds true.

$M$ is called log-convex, denoted by $(\text{lc})$, if

$\forall\;j\in\mathbb{N}_{>0}:\;M_{j}^{2}\leq M_{j-1}M_{j+1},$

equivalently if $\mu$ is nondecreasing. If $M$ is log-convex and normalized,
then both $M$ and $j\mapsto(M_{j})^{1/j}$ are nondecreasing. In this case we
get $M_{j}\geq 1$ for all $j\geq 0$ and

$\forall\;j\in\mathbb{N}_{>0}:\;\;\;(M_{j})^{1/j}\leq\mu_{j}.$

Moreover we get $M_{j}M_{k}\leq M_{j+k}$ for all $j,k\in\mathbb{N}$; e.g. see
[26, Lemma 2.0.4, Lemma 2.0.6].

If $m$ is log-convex, then $M$ is also log-convex and in this case we call $M$
strongly log-convex and write that $M$ is $(\text{slc})$.

For any $M=(M_{j})_{j}\in\mathbb{R}_{>0}^{\mathbb{N}}$ it is well-known that

(2.1)
$\liminf_{j\rightarrow\infty}\mu_{j}\leq\liminf_{j\rightarrow\infty}(M_{j})^{1/j}\leq\limsup_{j\rightarrow\infty}(M_{j})^{1/j}\leq\limsup_{j\rightarrow\infty}\mu_{j}.$

For convenience we introduce the following set of sequences:

$\hypertarget{LCset}{\mathcal{LC}}:=\\{M\in\mathbb{R}_{>0}^{\mathbb{N}}:\;M\;\text{is
normalized, log-convex},\;\lim_{j\rightarrow\infty}(M_{j})^{1/j}=\infty\\}.$

$M$ has moderate growth, denoted by $(\text{mg})$, if

$\exists\;C\geq 1\;\forall\;j,k\in\mathbb{N}:\;M_{j+k}\leq
C^{j+k+1}M_{j}M_{k}.$

A weaker condition is derivation closedness, denoted by $(\text{dc})$, if

$\exists\;A\geq 1\;\forall\;j\in\mathbb{N}:\;M_{j+1}\leq
A^{j+1}M_{j}\Leftrightarrow\mu_{j+1}\leq A^{j+1}.$

$M$ is called non-quasianalytic, denoted by $(\text{nq})$, if

$\sum_{j\geq 1}\frac{1}{\mu_{j}}<+\infty.$

In the literature $(\operatorname{mg})$ is also known under stability of
ultradifferential operators or $(M.2)$, $(\operatorname{dc})$ under
$(M.2)^{\prime}$ and $(\operatorname{nq})$ under $(M.3)^{\prime}$; see [11].
It is also known that for log-convex (normalized) weight sequences
$(\operatorname{nq})$ is equivalent to

$\sum_{j\geq 1}\frac{1}{(M_{j})^{1/j}}<+\infty,$

which holds by the so-called Carleman-inequality; see [26, Prop. 4.1.7] and
the references therein.

$M$ has $(\beta_{1})$ (named after [18]) if

$\exists\;Q\in\mathbb{N}_{>0}:\;\liminf_{j\rightarrow\infty}\frac{\mu_{Qj}}{\mu_{j}}>Q,$

and $(\gamma_{1})$ if

$\sup_{j\in\mathbb{N}_{>0}}\frac{\mu_{j}}{j}\sum_{k\geq
j}\frac{1}{\mu_{k}}<\infty.$

In [18, Proposition 1.1] it has been shown that for
$M\in\hyperlink{LCset}{\mathcal{LC}}$ both conditions are equivalent and in
the literature $(\gamma_{1})$ is also called “strong nonquasianalyticity
condition”. In [11] it is denoted by $(M.3)$ (in fact, there
$\frac{\mu_{j}}{j}$ is replaced by $\frac{\mu_{j}}{j-1}$ for $j\geq 2$ but
which is equivalent to having $(\gamma_{1})$).

A weaker condition on $M$ is $(\beta_{3})$ (named after [27], see also [2])
which reads as follows:

$\exists\;Q\in\mathbb{N}_{>0}:\;\liminf_{j\rightarrow\infty}\frac{\mu_{Qj}}{\mu_{j}}>1.$

For two weight sequences $M=(M_{j})_{j}$ and $N=(N_{j})_{j}$ we write
$M\hypertarget{mpreceq}{\preceq}N$ if

$\sup_{j\in\mathbb{N}_{>0}}\left(\frac{M_{j}}{N_{j}}\right)^{1/j}<\infty,$

and call them equivalent, denoted by $M\hypertarget{approx}{\approx}N$, if

$M\hyperlink{mpreceq}{\preceq}N\;\text{and}\;N\hyperlink{mpreceq}{\preceq}M.$

In the relations above one can replace $M$ and $N$ simultaneously by $m$ and
$n$ because $M\hyperlink{mpreceq}{\preceq}N\Leftrightarrow
m\hyperlink{mpreceq}{\preceq}n$. Let us also write $M\leq N$ if $M_{j}\leq
N_{j}$ for all $j\in\mathbb{N}$. Finally, we write
$M\hypertarget{triangle}{\vartriangleleft}N$, if

$\lim_{j\rightarrow\infty}\left(\frac{M_{j}}{N_{j}}\right)^{1/j}=0.$

For any $s\geq 0$ we set $G^{s}:=(j!^{s})_{j\in\mathbb{N}}$, so for $s>0$ this
denotes the classical Gevrey sequence of index/order $s$.

###  2.3. Weight functions

According to [16, Sect. 2.1] and [17, Sect. 2.2] a function
$\omega:[0,+\infty)\rightarrow[0,+\infty)$ is called a pre-weight function, if
it is continuous, non-decreasing, $\omega(0)=0$ and such that

  * $(*)$

$\log(t)=o(\omega(t))$, $t\rightarrow+\infty$,

  * $(*)$

$t\mapsto\varphi_{\omega}(t):=\omega(e^{t})$ is convex.

Consequently, for each pre-weight function we have
$\lim_{t\rightarrow+\infty}\omega(t)=+\infty$. $\omega$ is called a weight
function if $\omega$ satisfies in addition

  * $(*)$

$\omega(2t)=O(\omega(t))$, $t\rightarrow+\infty$.

If $\omega(t)=0$ for all $t\in[0,1]$, then we call $\omega$ a normalized
(pre-)weight function.

Let $\sigma,\tau$ be pre-weight functions, we write
$\sigma\hypertarget{ompreceq}{\preceq}\tau$ if
$\tau(t)=O(\sigma(t))\;\text{as}\;t\rightarrow+\infty$. We call them
equivalent, denoted by $\sigma\hypertarget{sim}{\sim}\tau$, if
$\sigma\hyperlink{ompreceq}{\preceq}\tau$ and
$\tau\hyperlink{ompreceq}{\preceq}\sigma$.

###  2.4. Associated weight function

Let $M\in\mathbb{R}_{>0}^{\mathbb{N}}$ (with $M_{0}=1$), then the associated
function $\omega_{M}:\mathbb{R}_{\geq
0}\rightarrow\mathbb{R}\cup\\{+\infty\\}$ is defined by

$\omega_{M}(t):=\sup_{j\in\mathbb{N}}\log\left(\frac{t^{j}}{M_{j}}\right)\;\;\;\text{for}\;t\in\mathbb{R}_{>0},\hskip
30.0pt\omega_{M}(0):=0.$

For an abstract introduction of the associated function we refer to [13,
Chapitre I]; see also [11, Definition 3.1].

If $\liminf_{j\rightarrow+\infty}(M_{j})^{1/j}>0$, then $\omega_{M}(t)=0$ for
sufficiently small $t$, since
$\log\left(\frac{t^{j}}{M_{j}}\right)<0\Leftrightarrow t<(M_{j})^{1/j}$ holds
for all $j\in\mathbb{N}_{>0}$. (In particular, if $M_{j}\geq 1$ for all
$j\in\mathbb{N}$, then $\omega_{M}$ is vanishing on $[0,1]$.) Moreover, under
this assumption $t\mapsto\omega_{M}(t)$ is a continuous nondecreasing
function, which is convex in the variable $\log(t)$ and tends faster to
infinity than any $\log(t^{j})$, $j\geq 1$, as $t\rightarrow+\infty$.
$\lim_{j\rightarrow+\infty}(M_{j})^{1/j}=+\infty$ implies that
$\omega_{M}(t)<+\infty$ for each finite $t$ which shall be considered as a
basic assumption for defining $\omega_{M}$.

Summarizing, if $M\in\hyperlink{LCset}{\mathcal{LC}}$, then $\omega_{M}$ is a
normalized pre-weight function (e.g. see [9, Lemma 3.1]), however in general
$\omega_{M}(2t)=O(\omega_{M}(t))$ is not clear; see the recent
characterization [30, Thm. 3.1].

Finally, if $M\in\hyperlink{LCset}{\mathcal{LC}}$ (or even if $M$ is log-
convex with $M_{0}=1$ and $\lim_{j\rightarrow+\infty}(M_{j})^{1/j}=+\infty$),
then by [13, Chapitre I, 1.4, 1.8] and also [11, Prop. 3.2] we get

(2.2) $M_{j}=\sup_{t\geq
0}\frac{t^{j}}{\exp(\omega_{M}(t))},\;\;\;j\in\mathbb{N}.$

###  2.5. Weight matrices

For the following definitions and conditions see [23, Sect. 4] and [27, Sect.
7].

Let $\mathcal{I}=\mathbb{R}_{>0}$ denote the index set (equipped with the
natural order). A weight matrix $\mathcal{M}$ associated with $\mathcal{I}$ is
a (one parameter) family of weight sequences
$\mathcal{M}:=\\{M^{(\alpha)}\in\mathbb{R}_{>0}^{\mathbb{N}}:\alpha\in\mathcal{I}\\}$,
such that

$\forall\;\alpha\leq\beta:\;\;\;M^{(\alpha)}\leq M^{(\beta)}.$

We call a weight matrix $\mathcal{M}$ standard log-convex, denoted by
$(\mathcal{M}_{\operatorname{sc}})$, if

$\forall\;\alpha\in\mathcal{I}:\;M^{(\alpha)}\in\hyperlink{LCset}{\mathcal{LC}}.$

Moreover, we put $m^{(\alpha)}_{j}:=\frac{M^{(\alpha)}_{j}}{j!}$ for
$j\in\mathbb{N}$, and
$\mu^{(\alpha)}_{j}:=\frac{M^{(\alpha)}_{j}}{M^{(\alpha)}_{j-1}}$ for
$j\in\mathbb{N}_{>0}$, $\mu^{(\alpha)}_{0}:=1$.

A matrix is called constant if
$M^{(\alpha)}\hyperlink{approx}{\approx}M^{(\beta)}$ for all
$\alpha,\beta\in\mathcal{I}$.

Let $\mathcal{M}=\\{M^{(\alpha)}:\alpha\in\mathcal{I}\\}$ and
$\mathcal{N}=\\{N^{(\alpha)}:\alpha\in\mathcal{I}\\}$ be given. We write
$\mathcal{M}\hypertarget{Mroumpreceq}{\\{\preceq\\}}\mathcal{N}$ if

$\forall\;\alpha\in\mathcal{I}\;\exists\;\beta\in\mathcal{I}:\;\;\;M^{(\alpha)}\hyperlink{preceq}{\preceq}N^{(\beta)},$

and call $\mathcal{M}$ and $\mathcal{N}$ to be $R$-equivalent, or
$\mathcal{M}\\{\approx\\}\mathcal{N}$ for short, if
$\mathcal{M}\hyperlink{Mroumpreceq}{\\{\preceq\\}}\mathcal{N}$ and
$\mathcal{N}\hyperlink{Mroumpreceq}{\\{\preceq\\}}\mathcal{M}$. Analogously,
we write $\mathcal{M}\hypertarget{Mbeurpreceq}{(\preceq)}\mathcal{N}$ if

$\forall\;\alpha\in\mathcal{I}\;\exists\;\beta\in\mathcal{I}:\;\;\;M^{(\beta)}\hyperlink{preceq}{\preceq}N^{(\alpha)},$

and call $\mathcal{M}$ and $\mathcal{N}$ to be $B$-equivalent, or
$\mathcal{M}(\approx)\mathcal{N}$ for short, if
$\mathcal{M}\hyperlink{Mbeurpreceq}{(\preceq)}\mathcal{N}$ and
$\mathcal{N}\hyperlink{Mbeurpreceq}{(\preceq)}\mathcal{M}$.

If $\mathcal{M}$ and $\mathcal{N}$ are both $R$\- and $B$-equivalent, then we
say for simplicity that they are equivalent.

We recall several growth and regularity assumptions on a given weight matrix:

$\displaystyle\hypertarget{holom}{(\mathcal{M}_{\mathcal{H}})}$
$\displaystyle\forall\;\alpha\in\mathcal{I}:\;\;\;\liminf_{j\rightarrow\infty}(m^{(\alpha)}_{j})^{1/j}>0,$
$\displaystyle\hypertarget{R-Comega}{(\mathcal{M}_{\\{\text{C}^{\omega}\\}})}$
$\displaystyle\exists\;\alpha\in\mathcal{I}:\;\;\;\liminf_{j\rightarrow\infty}(m^{(\alpha)}_{j})^{1/j}>0,$
$\displaystyle\hypertarget{B-Comega}{(\mathcal{M}_{(\text{C}^{\omega})})}$
$\displaystyle\forall\;\alpha\in\mathcal{I}:\;\;\;\lim_{j\rightarrow\infty}(m^{(\alpha)}_{j})^{1/j}=+\infty,$
$\displaystyle\hypertarget{R-rai}{(\mathcal{M}_{\\{\text{rai}\\}})}$
$\displaystyle\forall\;\alpha\in\mathcal{I}\;\exists\;C>0\;\exists\;\beta\in\mathcal{I}\;\forall\;1\leq
j\leq k:\;\;\;(m^{(\alpha)}_{j})^{1/j}\leq C(m^{(\beta)}_{k})^{1/k},$
$\displaystyle\hypertarget{B-rai}{(\mathcal{M}_{(\text{rai})})}$
$\displaystyle\forall\;\alpha\in\mathcal{I}\;\exists\;C>0\;\exists\;\beta\in\mathcal{I}\;\forall\;1\leq
j\leq k:\;\;\;(m^{(\beta)}_{j})^{1/j}\leq C(m^{(\alpha)}_{k})^{1/k},$
$\displaystyle\hypertarget{R-mg}{(\mathcal{M}_{\\{\text{mg}\\}})}$
$\displaystyle\forall\;\alpha\in\mathcal{I}\;\exists\;C>0\;\exists\;\beta\in\mathcal{I}\;\forall\;j,k\in\mathbb{N}:\;\;\;M^{(\alpha)}_{j+k}\leq
C^{j+k+1}M^{(\beta)}_{j}M^{(\beta)}_{k},$
$\displaystyle\hypertarget{B-mg}{(\mathcal{M}_{(\text{mg})})}$
$\displaystyle\forall\;\alpha\in\mathcal{I}\;\exists\;C>0\;\exists\;\beta\in\mathcal{I}\;\forall\;j,k\in\mathbb{N}:\;\;\;M^{(\beta)}_{j+k}\leq
C^{j+k+1}M^{(\alpha)}_{j}M^{(\alpha)}_{k},$
$\displaystyle\hypertarget{R-dc}{(\mathcal{M}_{\\{\text{dc}\\}})}$
$\displaystyle\forall\;\alpha\in\mathcal{I}\;\exists\;C>0\;\exists\;\beta\in\mathcal{I}\;\forall\;j\in\mathbb{N}:\;\;\;M^{(\alpha)}_{j+1}\leq
C^{j+1}M^{(\beta)}_{j},$
$\displaystyle\hypertarget{B-dc}{(\mathcal{M}_{(\text{dc})})}$
$\displaystyle\forall\;\alpha\in\mathcal{I}\;\exists\;C>0\;\exists\;\beta\in\mathcal{I}\;\forall\;j\in\mathbb{N}:\;\;\;M^{(\beta)}_{j+1}\leq
C^{j+1}M^{(\alpha)}_{j},$
$\displaystyle\hypertarget{R-BR}{(\mathcal{M}_{\\{\text{BR}\\}})}$
$\displaystyle\forall\;\alpha\in\mathcal{I}\;\exists\;\beta\in\mathcal{I}:\;\;\;M^{(\alpha)}\hyperlink{triangle}{\vartriangleleft}M^{(\beta)},$
$\displaystyle\hypertarget{B-BR}{(\mathcal{M}_{(\text{BR})})}$
$\displaystyle\forall\;\alpha\in\mathcal{I}\;\exists\;\beta\in\mathcal{I}:\;\;\;M^{(\beta)}\hyperlink{triangle}{\vartriangleleft}M^{(\alpha)},$
$\displaystyle\hypertarget{R-FdB}{(\mathcal{M}_{\\{\text{FdB}\\}})}$
$\displaystyle\forall\;\alpha\in\mathcal{I}\;\exists\;\beta\in\mathcal{I}:\;\;\;(m^{(\alpha)})^{\circ}\hyperlink{mpreceq}{\preceq}m^{(\beta)},$
$\displaystyle\hypertarget{B-FdB}{(\mathcal{M}_{(\text{FdB})})}$
$\displaystyle\forall\;\alpha\in\mathcal{I}\;\exists\;\beta\in\mathcal{I}:\;\;\;(m^{(\beta)})^{\circ}\hyperlink{mpreceq}{\preceq}m^{(\alpha)},$

with $(m^{(\alpha)})^{\circ}:=((m^{(\alpha)}_{j})^{\circ})_{j}$ being the
sequence defined by

$(m^{(\alpha)}_{k})^{\circ}:=\max\left\\{m^{(\alpha)}_{\ell}\cdot
m^{(\alpha)}_{j_{1}}\cdots
m^{(\alpha)}_{j_{\ell}}:j_{i}\in\mathbb{N}_{>0},\sum_{i=1}^{\ell}j_{i}=k\right\\},\;\;\;\;\;(m^{(\alpha)}_{0})^{\circ}:=1.$

$R$-equivalence between matrices preserves all Roumieu-type conditions listed
above and $B$-equivalence all Beurling-type conditions.

Finally, let us recall

$\displaystyle\hypertarget{R-L}{(\mathcal{M}_{\\{\text{L}\\}})}$
$\displaystyle\forall\;C>0\;\forall\;\alpha\in\mathcal{I}\;\exists\;D>0\;\exists\;\beta\in\mathcal{I}\;\forall\;j\in\mathbb{N}:C^{j}M^{(\alpha)}_{j}\leq
DM^{(\beta)}_{j},$
$\displaystyle\hypertarget{B-L}{(\mathcal{M}_{(\text{L})})}$
$\displaystyle\forall\;C>0\;\forall\;\alpha\in\mathcal{I}\;\exists\;D>0\;\exists\;\beta\in\mathcal{I}\;\forall\;j\in\mathbb{N}:C^{j}M^{(\beta)}_{j}\leq
DM^{(\alpha)}_{j}.$

A matrix is called non-quasianalytic if any sequence $M^{(\alpha)}$ is non-
quasianalytic; see [29, Sect. 4]. When dealing with Roumieu type classes then
it suffices to assume that there exists $\alpha_{0}\in\mathcal{I}$ such that
$M^{(\alpha_{0})}$ is non-quasianalytic since smaller indices can be skipped;
see also the discussion in [25, Sect. 5.1].

###  2.6. Ultradifferentiable classes

Let $U\subseteq\mathbb{R}^{d}$ be non-empty open and for
$K\subseteq\mathbb{R}^{d}$ compact we write $K\subset\subset U$ if
$\overline{K}\subseteq U$, i.e., $K$ is in $U$ relatively compact. We
introduce now the following spaces of ultradifferentiable function classes.
First, for weight sequences we define the (local) classes of Roumieu-type by

$\mathcal{E}_{\\{M\\}}(U):=\\{f\in\mathcal{E}(U):\;\forall\;K\subset\subset
U\;\exists\;h>0:\;\|f\|_{M,K,h}<+\infty\\},$

and the classes of Beurling-type by

$\mathcal{E}_{(M)}(U):=\\{f\in\mathcal{E}(U):\;\forall\;K\subset\subset
U\;\forall\;h>0:\;\|f\|_{M,K,h}<+\infty\\},$

where we denote

$\|f\|_{M,K,h}:=\sup_{\alpha\in\mathbb{N}^{d},x\in
K}\frac{|f^{(\alpha)}(x)|}{h^{|\alpha|}M_{|\alpha|}}.$

For a sufficiently regular compact set $K$ (e.g. with smooth boundary and such
that $\overline{K^{\circ}}=K$)

$\mathcal{E}_{M,h}(K):=\\{f\in\mathcal{E}(K):\|f\|_{M,K,h}<+\infty\\}$

is a Banach space and so we have the following topological vector spaces

$\mathcal{E}_{\\{M\\}}(K):=\underset{h>0}{\varinjlim}\;\mathcal{E}_{M,h}(K),$

and

$\mathcal{E}_{\\{M\\}}(U)=\underset{K\subset\subset
U}{\varprojlim}\;\underset{h>0}{\varinjlim}\;\mathcal{E}_{M,h}(K)=\underset{K\subset\subset
U}{\varprojlim}\;\mathcal{E}_{\\{M\\}}(K).$

Similarly, we get

$\mathcal{E}_{(M)}(K):=\underset{h>0}{\varprojlim}\;\mathcal{E}_{M,h}(K),$

and

$\mathcal{E}_{(M)}(U)=\underset{K\subset\subset
U}{\varprojlim}\;\underset{h>0}{\varprojlim}\;\mathcal{E}_{M,h}(K)=\underset{K\subset\subset
U}{\varprojlim}\;\mathcal{E}_{(M)}(K).$

For a weight function, we define the corresponding Roumieu-type classes

$\mathcal{E}_{\\{\omega\\}}(U):=\\{f\in\mathcal{E}(U):\;\forall\;K\subset\subset
U\;\exists\;h>0:\;\|f\|_{\omega,K,h}<+\infty\\},$

and the classes of Beurling-type by

$\mathcal{E}_{(\omega)}(U):=\\{f\in\mathcal{E}(U):\;\forall\;K\subset\subset
U\;\forall\;h>0:\;\|f\|_{\omega,K,h}<+\infty\\},$

where we denote

$\|f\|_{\omega,K,h}:=\sup_{\alpha\in\mathbb{N}^{d},x\in
K}\frac{|f^{(\alpha)}(x)|}{\exp(\frac{1}{h}\varphi_{\omega}^{*}(h|\alpha|))}.$

The spaces are topologized in complete analogy to the weight sequence case.
First we define for a sufficiently regular compact set $K$ the Banach space

$\mathcal{E}_{\omega,h}(K):=\\{f\in\mathcal{E}(K):\|f\|_{\omega,K,h}<+\infty\\},$

and set

$\mathcal{E}_{\\{\omega\\}}(K):=\underset{h>0}{\varinjlim}\;\mathcal{E}_{\omega,h}(K),\quad\mathcal{E}_{(\omega)}(K):=\underset{h>0}{\varprojlim}\;\mathcal{E}_{\omega,h}(K),$

finally we endow $\mathcal{E}_{[\omega]}(U)$ with the following locally convex
topologies

$\mathcal{E}_{\\{\omega\\}}(U)=\underset{K\subset\subset
U}{\varprojlim}\;\mathcal{E}_{\\{\omega\\}}(K),\quad\mathcal{E}_{(\omega)}(U)=\underset{K\subset\subset
U}{\varprojlim}\;\mathcal{E}_{(\omega)}(K).$

Next, we consider classes defined by weight matrices of Roumieu-type
$\mathcal{E}_{\\{\mathcal{M}\\}}$ and of Beurling-type
$\mathcal{E}_{(\mathcal{M})}$ as follows; see also [23, 4.2]. For a weight
matrix $\mathcal{M}=\\{M^{(x)}:x\in\mathcal{I}\\}$ and a sufficiently regular
$K\subset\subset U$ we put

(2.3)
$\mathcal{E}_{\\{\mathcal{M}\\}}(K):=\bigcup_{x\in\mathcal{I}}\mathcal{E}_{\\{M^{(x)}\\}}(K),\hskip
20.0pt\mathcal{E}_{\\{\mathcal{M}\\}}(U):=\bigcap_{K\subset\subset
U}\bigcup_{x\in\mathcal{I}}\mathcal{E}_{\\{M^{(x)}\\}}(K),$

and

$\mathcal{E}_{(\mathcal{M})}(K):=\bigcap_{x\in\mathcal{I}}\mathcal{E}_{(M^{(x)})}(K),\hskip
20.0pt\mathcal{E}_{(\mathcal{M})}(U):=\bigcap_{x\in\mathcal{I}}\mathcal{E}_{(M^{(x)})}(U).$

For such $K$ one has the representation

$\mathcal{E}_{\\{\mathcal{M}\\}}(K)=\underset{x\in\mathcal{I}}{\varinjlim}\;\underset{h>0}{\varinjlim}\;\mathcal{E}_{M^{(x)},h}(K)$

and so for $U\subseteq\mathbb{R}^{d}$ non-empty open

$\mathcal{E}_{\\{\mathcal{M}\\}}(U)=\underset{K\subset\subset
U}{\varprojlim}\;\underset{x\in\mathcal{I}}{\varinjlim}\;\underset{h>0}{\varinjlim}\;\mathcal{E}_{M^{(x)},h}(K).$

Similarly, we get for the Beurling case

$\mathcal{E}_{(\mathcal{M})}(U)=\underset{K\subset\subset
U}{\varprojlim}\;\underset{x\in\mathcal{I}}{\varprojlim}\;\underset{h>0}{\varprojlim}\;\mathcal{E}_{M^{(x)},h}(K).$

###  2.7. Pilipović-Teofanov-Tomić classes

Let $\tau,h>0$, $\sigma\geq 1$, then one considers the weight sequence

$M^{\tau,\sigma}_{j}:=j^{\tau j^{\sigma}}.$

Let $K\subset\subset\mathbb{R}^{d}$ be a sufficiently regular compact set. By
$\mathcal{E}_{\tau,\sigma,h}(K)$ we shall denote the Banach space of functions
$\phi\in\mathcal{E}(K)$ such that

$||\phi||_{\mathcal{E}_{\tau,\sigma,h}(K)}=\sup_{\alpha\in\mathbb{N}^{d}}\sup_{x\in
K}\frac{|\phi^{(\alpha)}(x)|}{h^{|\alpha|^{\sigma}}M_{|\alpha|}^{\tau,\sigma}}<+\infty.$

Note that the case $\sigma=1$ gives, by Stirling’s formula, that
$M^{\tau,1}\hyperlink{approx}{\approx}G^{\tau}$, i.e., $M^{\tau,1}$ is
equivalent to the classical Gevrey sequence with parameter/index $\tau$.

If $0<h_{1}\leq h_{2}$, $0<\tau_{1}\leq\tau_{2}$,
$1\leq\sigma_{1}\leq\sigma_{2}$, then
$\mathcal{E}_{\tau_{1},\sigma_{1},h_{1}}(K)\hookrightarrow\mathcal{E}_{\tau_{2},\sigma_{2},h_{2}}(K)$.

Let $U$ be an open set of $\mathbb{R}^{d}$. We define the spaces:

$\mathcal{E}_{\\{\tau,\sigma\\}}(U)=\varprojlim_{K\subset\subset
U}\varinjlim_{h\to\infty}\mathcal{E}_{\tau,\sigma,h}(K)\qquad\qquad\mathcal{E}_{(\tau,\sigma)}(U)=\varprojlim_{K\subset\subset
U}\varprojlim_{h\to 0}\mathcal{E}_{\tau,\sigma,h}(K)$
$\mathcal{E}_{\\{\infty,\sigma\\}}(U)=\varinjlim_{\tau\to\infty}\mathcal{E}_{\\{\tau,\sigma\\}}(U)\qquad\qquad\mathcal{E}_{(\infty,\sigma)}(U)=\varinjlim_{\tau\to\infty}\mathcal{E}_{(\tau,\sigma)}(U)$
$\mathcal{E}_{\\{0,\sigma\\}}(U)=\varprojlim_{\tau\to
0}\mathcal{E}_{\\{\tau,\sigma\\}}(U)\qquad\qquad\mathcal{E}_{(0,\sigma)}(U)=\varprojlim_{\tau\to
0}\mathcal{E}_{(\tau,\sigma)}(U)$

We use the abbreviated notation $[\tau,\sigma]$ for $\\{\tau,\sigma\\}$ or
$(\tau,\sigma)$.

######  Proposition 2.1.

[19, Prop. 2.1] [20, Prop. 2.1] Let $\sigma\geq 1$ and $\tau>0$. Then for
every $\sigma_{2}>\sigma_{1}\geq 1$, we have that

$\mathcal{E}_{[\infty,\sigma_{1}]}(U)\hookrightarrow\mathcal{E}_{[0,\sigma_{2}]}(U).$

Moreover, if $0<\tau_{1}<\tau_{2}$ then

$\mathcal{E}_{\\{\tau_{1},\sigma\\}}(U)\hookrightarrow\mathcal{E}_{(\tau_{2},\sigma)}(U)\hookrightarrow\mathcal{E}_{\\{\tau_{2},\sigma\\}}(U),$

and

$\mathcal{E}_{\\{\infty,\sigma\\}}(U)=\mathcal{E}_{(\infty,\sigma)}(U)\qquad\qquad\mathcal{E}_{\\{0,\sigma\\}}(U)=\mathcal{E}_{(0,\sigma)}(U).$

Consequently, given $\tau_{0}>0$ we see that

$\mathcal{E}_{[\tau_{0},\sigma_{1}]}(U)\hookrightarrow\bigcap_{\tau>\tau_{0}}\mathcal{E}_{[\tau,\sigma_{1}]}(U)\hookrightarrow\mathcal{E}_{[\tau_{0},\sigma_{2}]}(U).$

In particular, if $\sigma>1$ then

$\mathcal{E}_{[\infty,1]}(U)\hookrightarrow\mathcal{E}_{[\tau,\sigma]}(U).$

##  3\. Ultradifferentiable classes beyond geometric growth factors

The main objective is to prove that the PTT-classes can be represented as
classes defined by (suitable) weight matrices. Indeed, we can obtain a more
general result by letting the exponents of the defining estimates be
$\Phi_{j}$ instead of ${j}$ or $j^{\sigma}$ where $\Phi=(\Phi_{j})_{j}$ is
arbitrary and only satisfying some mild regularity property.

######  Definition 3.1.

A sequence $\Phi\in\mathbb{R}_{\geq 0}^{\mathbb{N}}$ is called _exponent
sequence_ if it satisfies

(3.1) $\liminf_{j\rightarrow\infty}\frac{\Phi_{j}}{j}>0.$

In particular every exponent sequence tends to infinity.

Let $U\subseteq\mathbb{R}^{d}$ be non-empty open,
$M\in\mathbb{R}_{>0}^{\mathbb{N}}$ and $\Phi\in\mathbb{R}_{\geq
0}^{\mathbb{N}}$. We introduce now $\Phi$-ultradifferentiable function classes
(defined in terms of a single weight sequence $M$): The (local) class of
Roumieu-type is given by

$\mathcal{E}_{\\{M,\Phi\\}}(U):=\\{f\in\mathcal{E}(U):\;\forall\;K\subset\subset
U\;\exists\;h>0:\;\|f\|_{M,\Phi,K,h}<+\infty\\},$

and the class of Beurling-type by

$\mathcal{E}_{(M,\Phi)}(U):=\\{f\in\mathcal{E}(U):\;\forall\;K\subset\subset
U\;\forall\;h>0:\;\|f\|_{M,\Phi,K,h}<+\infty\\},$

where we denote

(3.2) $\|f\|_{M,\Phi,K,h}:=\sup_{\alpha\in\mathbb{N}^{d},x\in
K}\frac{|f^{(\alpha)}(x)|}{h^{\Phi_{|\alpha|}}M_{|\alpha|}}.$

For a sufficiently regular compact set $K$

$\mathcal{E}_{M,\Phi,h}(K):=\\{f\in\mathcal{E}(K):\|f\|_{M,\Phi,K,h}<+\infty\\}$

is a Banach space and so we have the following topological vector space
representations

$\mathcal{E}_{\\{M,\Phi\\}}(K):=\underset{h>0}{\varinjlim}\;\mathcal{E}_{M,\Phi,h}(K),\qquad\text{and}\qquad\mathcal{E}_{\\{M,\Phi\\}}(U)=\underset{K\subset\subset
U}{\varprojlim}\;\underset{h>0}{\varinjlim}\;\mathcal{E}_{M,\Phi,h}(K)=\underset{K\subset\subset
U}{\varprojlim}\;\mathcal{E}_{\\{M,\Phi\\}}(K).$

Similarly, we get

$\mathcal{E}_{(M,\Phi)}(K):=\underset{h>0}{\varprojlim}\;\mathcal{E}_{M,\Phi,h}(K),\qquad\text{and}\qquad\mathcal{E}_{(M,\Phi)}(U)=\underset{K\subset\subset
U}{\varprojlim}\;\underset{h>0}{\varprojlim}\;\mathcal{E}_{M,\Phi,h}(K)=\underset{K\subset\subset
U}{\varprojlim}\;\mathcal{E}_{(M,\Phi)}(K).$

Note that condition (3.1) means that the factor $h^{\Phi_{|\alpha|}}$ in the
seminorms is at least geometric.

######  Remark 3.2.

In the literature, global ultradifferentiable classes, test function spaces,
ultraholomorphic classes, spaces of weighted sequences of complex numbers and,
PTT-test function spaces (see [19]) have been studied. In a completely
analogous way, global $\Phi$-ultradifferentiable classes, $\Phi$-test function
spaces, $\Phi$-ultraholomorphic classes and $\Phi$-spaces of weighted
sequences of complex numbers (weighted with seminorms of the type (3.2)) can
be defined ; i.e., when the symbol/functor $\mathcal{E}$ is replaced by
$\mathcal{B}$, $\mathcal{D}$, $\mathcal{A}$ or $\Lambda$.

Notation: When the open set $U$ is not relevant in certain statement, we will
simply write $\mathcal{E}_{\\{\mathcal{M}\\}}$, $\mathcal{E}_{(\mathcal{M})}$,
$\mathcal{E}_{\\{\tau,\sigma\\}}$, $\mathcal{E}_{(\tau,\sigma)}$ or
$\mathcal{E}_{\\{M,\Phi\\}}$, $\mathcal{E}_{(M,\Phi)}$.

######  Example 3.3.

We have two important examples in mind:

  * $(a)$

If $\Phi_{j}=j$ for all $j\in\mathbb{N}$, then we recover the classes from
Section 2.6; i.e., the usual definition of ultradifferentiable classes defined
in terms of weight sequences; e.g. see [11]. For reasons of simplicity we will
skip the letter $\Phi$ in the definition.

  * $(b)$

Let parameters $\sigma>1$ and $\tau>0$ be given. Then the choices

(3.3) $M^{\tau,\sigma}_{j}:=j^{\tau j^{\sigma}},\hskip
15.0pt\Phi_{j}:=j^{\sigma},\;\;\;j\in\mathbb{N},$

yield the classes introduced in [19]. We use the convention $0^{0}:=1$ and so
$M^{\tau,\sigma}_{0}=1$.

Given an exponent sequence $\Phi\in\mathbb{R}_{\geq 0}^{\mathbb{N}}$ and a
pair of sequences $M,N\in\mathbb{R}_{>0}^{\mathbb{N}}$, we want to compare the
classes $\mathcal{E}_{\\{M,\Phi\\}}$ and $\mathcal{E}_{\\{N,\Phi\\}}$, or
resp. $\mathcal{E}_{(M,\Phi)}$ and $\mathcal{E}_{(N,\Phi)}$. We write
$M\hypertarget{preceq}{\preceq_{\Phi}}N$ if

$\sup_{j\in\mathbb{N}_{>0}}\left(\frac{M_{j}}{N_{j}}\right)^{1/\Phi_{j}}<+\infty,$

and we call $M$ and $N$ to be $\Phi$-equivalent, written as
$M\hypertarget{approx}{\approx_{\Phi}}N$, if
$M\hyperlink{preceq}{\preceq_{\Phi}}N$ and
$N\hyperlink{preceq}{\preceq_{\Phi}}M$.

By definition obviously $M\hyperlink{preceq}{\preceq_{\Phi}}N$ implies both
$\mathcal{E}_{\\{M,\Phi\\}}\subseteq\mathcal{E}_{\\{N,\Phi\\}}$ and
$\mathcal{E}_{(M,\Phi)}\subseteq\mathcal{E}_{(N,\Phi)}$ with continuous
inclusion. Thus $\Phi$-equivalent sequences define the same associated
function classes; i.e., if $M\hyperlink{approx}{\approx_{\Phi}}N$ then
$\mathcal{E}_{\\{M,\Phi\\}}=\mathcal{E}_{\\{N,\Phi\\}}$ and
$\mathcal{E}_{(M,\Phi)}=\mathcal{E}_{(N,\Phi)}$ (as topological vector spaces)
and similarly for the other classes mentioned in Remark 3.2. Note that for
$\Phi_{j}=j$ relation $\preceq_{\Phi}$ is precisely $\preceq$; see [23, p.
101, Prop. 2.12 $(1)$].

Analogously, we write $M\hypertarget{triangle}{\vartriangleleft_{\Phi}}N$ if

$\lim_{j\rightarrow+\infty}\left(\frac{M_{j}}{N_{j}}\right)^{1/\Phi_{j}}=0,$

which is obviously never reflexive and symmetric and stronger than
$\preceq_{\Phi}$. If $M\hyperlink{triangle}{\vartriangleleft_{\Phi}}N$, then
$\mathcal{E}_{\\{M,\Phi\\}}\subseteq\mathcal{E}_{(N,\Phi)}$ with continuous
inclusion (and similarly for the other classes mentioned in Remark 3.2).

##  4\. Classes beyond geometric factors versus weight matrices

The aim of this section is to verify that, under mild growth and regularity
assumptions on $\Phi$, the classes $\mathcal{E}_{\\{M,\Phi\\}}$ resp.
$\mathcal{E}_{(M,\Phi)}$ can be represented (as locally convex vector spaces)
by the matrix classes $\mathcal{E}_{\\{\mathcal{M}\\}}$ resp.
$\mathcal{E}_{(\mathcal{M})}$ for a suitable choice of the matrix
$\mathcal{M}$.

###  4.1. Preparatory results

We introduce an appropriate matrix of sequences. Let
$M\in\mathbb{R}_{>0}^{\mathbb{N}}$ and $\Phi\in\mathbb{R}_{\geq
0}^{\mathbb{N}}$, then consider the set

(4.1) $\mathcal{M}_{M,\Phi}:=\\{M^{(c,\Phi)}:c>0\\},\hskip
15.0ptM^{(c,\Phi)}_{j}:=c^{\Phi_{j}}M_{j},\;\;\;j\in\mathbb{N}.$

For any sequence $\Phi\in\mathbb{R}_{\geq 0}^{\mathbb{N}}$, we have that
$\mathcal{M}_{M,\Phi}$ is a weight matrix in the notion of Section 2.5. Next
we show that the mild growth restriction (3.1) on $\Phi$ is equivalent to the
fact that the matrix $\mathcal{M}_{M,\Phi}$ allows to absorb exponential
growth.

######  Lemma 4.1.

Let $M\in\mathbb{R}_{>0}^{\mathbb{N}}$ and $\Phi\in\mathbb{R}_{\geq
0}^{\mathbb{N}}$ be given. Then the following are equivalent:

  1. $(i)$

$\Phi$ satisfies (3.1), i.e., $\Phi$ is an exponent sequence.

  2. $(ii)$

$\mathcal{M}_{M,\Phi}$ satisfies $(\mathcal{M}_{\\{\operatorname{L}\\}})$.

  3. $(iii)$

$\mathcal{M}_{M,\Phi}$ satisfies $(\mathcal{M}_{(\operatorname{L})})$.

###### Proof.

$(i)\Leftrightarrow(ii)$ First we observe that $\mathcal{M}_{M,\Phi}$
satisfies $(\mathcal{M}_{\\{\text{L}\\}})$ if and only if

$\forall\;h>0\;\forall\;c>0\;\exists\;D>0\;\exists\;c_{1}>0\;\forall\;j\in\mathbb{N}:h^{j}c^{\Phi_{j}}\leq
Dc_{1}^{\Phi_{j}}.$

Consequently, if (ii) holds, then for $h=2$, $c=1$ there exists $c_{1}>1$ and
$D>0$ such that

$\frac{\Phi_{j}}{j}\geq\frac{\log(2)-(1/j)\log(D)}{\log(c_{1})},$

for all $j\in\mathbb{N}_{>0}$, so $\Phi$ satisfies (3.1).

Conversely, let $h,c\geq 1$ be given. Then (3.1) yields the existence of some
$\epsilon>0$ and $j_{\epsilon}\in\mathbb{N}_{>0}$ such that for all $j\geq
j_{\epsilon}$ we get $\frac{\Phi_{j}}{j}\geq\epsilon$. So there exists some
$c_{1}>c$ such that
$\left(\frac{c_{1}}{c}\right)^{\Phi_{j}/j}\geq\left(\frac{c_{1}}{c}\right)^{\epsilon}\geq
h$ for all $j\geq j_{\epsilon}$. Thus, when choosing $A\geq 1$ large enough,
then we have that for all $j\in\mathbb{N}$

$h^{j}M^{(c,\Phi)}_{j}=h^{j}c^{\Phi_{j}}M_{j}\leq
Ac_{1}^{\Phi_{j}}M_{j}=AM^{(c_{1},\Phi)}_{j},$

which shows that $\mathcal{M}_{M,\Phi}$ satisfies
$(\mathcal{M}_{\\{\text{L}\\}})$.

$(i)\Leftrightarrow(iii)$ Follows similarly. ∎

If we assume more growth requirements on the sequence $\Phi$, then we can
deduce further regularity conditions for $\mathcal{M}_{M,\Phi}$.

######  Proposition 4.2.

Let $M\in\mathbb{R}_{>0}^{\mathbb{N}}$ and $\Phi\in\mathbb{R}_{\geq
0}^{\mathbb{N}}$ be given.

  * $(i)$

If $M$ is normalized and if $\Phi_{0}=\Phi_{1}=0$, then each $M^{(c,\Phi)}$ is
normalized, too.

  * $(ii)$

We observe that $M^{(c,\Phi)}$ is log-convex if and only if

(4.2) $j\mapsto\mu^{(c,\Phi)}_{j}=c^{\Phi_{j}-\Phi_{j-1}}\mu_{j}\qquad\text{is
non-decreasing.}$

In particular, if $M$ is log-convex, i.e., $j\mapsto\mu_{j}$ is non-
decreasing, and if $\Phi$ is convex, i.e.,

$\forall\;j\in\mathbb{N}_{>0}:\;\;\;2\Phi_{j}\leq\Phi_{j-1}+\Phi_{j+1},$

then $M^{(c,\Phi)}$ is log-convex for each $c\geq 1$.

Moreover, if $\Phi$ is convex and (4.2) holds for some $c>0$, then also for
all $d>c$.

  * $(iii)$

If $\Phi$ is increasing, convex, and $\Phi_{0}=0$, then it is an exponent
sequence. Moreover, since $\Phi_{j}-\Phi_{j-1}\geq 0$, we get

$\forall\;0<c_{1}\leq c_{2}\;\forall\;j\geq
1:\;\;\;\mu^{(c_{1},\Phi)}_{j}=\frac{M^{(c_{1},\Phi)}_{j}}{M^{(c_{1},\Phi)}_{j-1}}=c_{1}^{\Phi_{j}-\Phi_{j-1}}\mu_{j}\leq
c_{2}^{\Phi_{j}-\Phi_{j-1}}\mu_{j}=\mu^{(c_{2},\Phi)}_{j},$

i.e., the sequences are even ordered w.r.t. their corresponding quotient
sequences.

  * $(iv)$

If $(M_{j})^{1/j}\rightarrow\infty$ as $j\rightarrow\infty$, and $\Phi$ is an
exponent sequence, then

$\forall\;c\geq
1:\;\;(M^{(c,\Phi)}_{j})^{1/j}=c^{\Phi_{j}/j}(M_{j})^{1/j}\rightarrow\infty,\;j\rightarrow\infty.$

###### Proof.

$(i)$ and $(iv)$ are direct consequences. For the last statement in $(ii)$ let
$d>c$ and write

$d^{\Phi_{j}-\Phi_{j-1}}\mu_{j}=c^{\Phi_{j}-\Phi_{j-1}}\mu_{j}\left(\frac{d}{c}\right)^{\Phi_{j}-\Phi_{j-1}}.$

Since $\frac{d}{c}\geq 1$ and the convexity of $\Phi$ precisely means that
$j\mapsto\Phi_{j}-\Phi_{j-1}$ is non-decreasing we get that $j\mapsto
d^{\Phi_{j}-\Phi_{j-1}}\mu_{j}$ is non-decreasing as well.

Let us give an argument for $(iii)$: From convexity since $\Phi_{0}=0$, one
can deduce that $j\mapsto\frac{\Phi_{j}}{j}$ is non-decreasing:

Observe that

$\Phi_{j}-\Phi_{j-1}\geq\frac{\Phi_{j-1}}{j-1},$

and therefore

$\Phi_{j}=(j-1)\frac{\Phi_{j-1}}{j-1}+\Phi_{j}-\Phi_{j-1}\geq
j\frac{\Phi_{j-1}}{j-1}.$

Since $\Phi$ is increasing we get $\Phi_{j}>0$ for all large $j$ and hence
$\frac{\Phi_{j}}{j}$ is bounded away from $0$. ∎

######  Remark 4.3.

Since log-convexity for $M^{(c,\Phi)}$ and
$(M^{(c,\Phi)}_{j})^{1/j}\rightarrow\infty$ as $j\rightarrow\infty$ for each
$c>0$ are desirable (standard) properties in the theory of ultradifferentiable
(and ultraholomorphic) classes, statements $(ii)$ and $(iv)$ in Proposition
4.2 suggest that for applications the choices for $\Phi$ and $M$ should not
considered to be independent; cf. (3.3).

In concrete applications the requirement
$(M^{(c,\Phi)}_{j})^{1/j}\rightarrow\infty$ as $j\rightarrow\infty$ for each
$c>0$ might be checked easily. However, even if $M$ is log-convex, $\Phi$ is
convex and $M$ and $\Phi$ are well related as in (3.3), in general as
$c\rightarrow 0$ one can only expect that condition (4.2) will be satisfied
from some $j_{c}\in\mathbb{N}_{>0}$ on (and $j_{c}\rightarrow+\infty$ as
$c\rightarrow 0$). Nevertheless, in this situation one can replace each
$M^{(c,\Phi)}$ (for $c<1$ small) by some equivalent sequence when changing
$M^{(c,\Phi)}$ at the beginning, i.e., only for finitely many $j$. This
technical modification leaves the classes
$\mathcal{E}_{\\{\mathcal{M}_{M,\Phi}\\}}$ and
$\mathcal{E}_{(\mathcal{M}_{M,\Phi})}$ unchanged.

###  4.2. Comparison results

This section is devoted to formulate and prove the main comparison theorems.
Using the preparation from the previous section we are in position to prove
the first statement.

######  Theorem 4.4.

Let $M\in\mathbb{R}_{>0}^{\mathbb{N}}$ be given and let $\Phi$ be an exponent
sequence. Let $\mathcal{M}_{M,\Phi}$ be the matrix defined in (4.1), then as
locally convex vector spaces we get

(4.3)
$\mathcal{E}_{\\{M,\Phi\\}}=\mathcal{E}_{\\{\mathcal{M}_{M,\Phi}\\}},\hskip
15.0pt\mathcal{E}_{(M,\Phi)}=\mathcal{E}_{(\mathcal{M}_{M,\Phi})}.$

By the analogous definitions of the spaces we see that (4.3) also holds for
the other classes mentioned in Remark 3.2.

Both cases from Example 3.3 satisfy (3.1); if $\Phi_{j}=j$ and so we are
treating the classical situation, then the above result becomes trivial in the
sense that $M^{(c,\Phi)}\hyperlink{approx}{\approx}M$ for all $c>0$, i.e.,
$\mathcal{M}_{M,\Phi}$ is constant.

Proof. The Roumieu case. By definition, we have the following estimate

$\forall\;c\geq 1\;\forall\;h\geq
1\;\forall\;j\in\mathbb{N}:\;\;\;c^{\Phi_{j}}M_{j}=M^{(c,\Phi)}_{j}\leq
h^{j}M^{(c,\Phi)}_{j},$

that is verifying
$\mathcal{E}_{\\{M,\Phi\\}}\subseteq\mathcal{E}_{\\{\mathcal{M}_{M,\Phi}\\}}$
(with continuous inclusion).

Conversely, let $h,c\geq 1$ be given. By Lemma 4.1, there exist $c_{1},A\geq
1$ such that for all $j\in\mathbb{N}$

$h^{j}M^{(c,\Phi)}_{j}\leq AM^{(c_{1},\Phi)}_{j},$

which shows
$\mathcal{E}_{\\{\mathcal{M}_{M,\Phi}\\}}\subseteq\mathcal{E}_{\\{M,\Phi\\}}$
(with continuous inclusion).

The Beurling case. Follows analogously, but in this case we use Lemma 4.1 to
prove the (continuous) inclusion
$\mathcal{E}_{(M,\Phi)}\subseteq\mathcal{E}_{(\mathcal{M}_{M,\Phi})}$ . ∎

On the other hand, let us show now that condition (3.1) is also necessary to
obtain (4.3) (or even more), when assuming mild extra assumptions on $M$.

A crucial part of the proof of the Roumieu case is based upon the existence of
so-called optimal functions in Roumieu classes: For any given normalized log-
convex sequence $N$, we consider the function

(4.4)
$\theta_{N}(t):=\sum_{j=0}^{\infty}\frac{N_{j}}{2^{j}\nu_{j}^{j}}\exp(2i\nu_{j}t),\;\;\;t\in\mathbb{R},$

with $\nu_{j}:=\frac{N_{j}}{N_{j-1}}$ for $j\geq 1$ and $\nu_{0}:=1$. It is
known that

$\theta_{N}\in\mathcal{E}_{\\{N\\}}(\mathbb{R},\mathbb{C}),\hskip
25.0pt|\theta_{N}^{(j)}(0)|\geq N_{j}\;\;\;\forall\;j\in\mathbb{N};$

see e.g. [35, Thm. 1], [23, Lemma 2.9] and the detailed proof in [26, Prop.
3.1.2]. There it has been commented that
$\theta_{N}\notin\mathcal{E}_{(N)}(\mathbb{R},\mathbb{C})$ and the proof shows
that in (4.4) we can replace $\nu_{j}$ by $\nu_{j+1}$.

The proof of the Beurling case makes use of the following functional analytic
result.

######  Proposition 4.5.

Let $E,F$ be Fréchet spaces, such that $E$ is a linear subspace of $F$ (not
assuming continuous inclusion). Assume that both are continuously included in
$C(U)$ (or even in any Hausdorff space), where $U$ is some open subset of
$\mathbb{R}^{d}$. Then $E$ is continuously included in $F$.

###### Proof.

We want to show that the inclusion $\iota:E\rightarrow F$ is continuous. By
the closed graph theorem, it suffices to show that if $f_{n}\rightarrow 0$ in
$E$ (and thus in $C(U)$), and $f_{n}\rightarrow g$ in $F$ (and thus in
$C(U)$), we have $g=0$. But this is now clear since $C(U)$ is Hausdorff. ∎

######  Remark 4.6.

In the light of the previous Proposition, any inclusion (as sets) of Beurling
classes is automatically a continuous inclusion. And equality as sets yields
equality as Fréchet spaces.

Now we are in the position to formulate a converse to Theorem 4.4.

######  Theorem 4.7.

Let $M\in\mathbb{R}_{>0}^{\mathbb{N}}$ and $\Phi\in\mathbb{R}_{\geq
0}^{\mathbb{N}}$ be given. Then we get:

  * $(i)$

The Roumieu case. Assume that $M^{(c,\Phi)}$ is log-convex and normalized for
some $c\geq 1$, and let $\mathcal{L}=\\{L^{(x)}:x>0\\}$ be a
$(\mathcal{M}_{\operatorname{sc}})$ matrix or even only consisting of
normalized log-convex weight sequences. Assume that

$\mathcal{E}_{\\{M,\Phi\\}}(\mathbb{R})=\mathcal{E}_{\\{\mathcal{L}\\}}(\mathbb{R})$

is valid (as sets). Then $\Phi$ has to satisfy (3.1), i.e., $\Phi$ is an
exponent sequence.

  * $(ii)$

The Beurling case. Assume that
$\lim_{j\rightarrow+\infty}(M^{(c,\Phi)}_{j})^{1/j}=+\infty$ for all $c>0$ and
such that (w.l.o.g., cf. Remark 4.3) each $M^{(c,\Phi)}$ is log-convex and
$M^{(c,\Phi)}_{0}=1$. Assume also that (as sets and thus automatically as
Fréchet spaces)

$\mathcal{E}_{(M,\Phi)}(\mathbb{R})=\mathcal{E}_{(\mathcal{L})}(\mathbb{R})$

is valid with $\mathcal{L}=\\{L^{(x)}:x>0\\}$ a given
$(\mathcal{M}_{\operatorname{sc}})$ matrix. Then $\Phi$ has to satisfy (3.1).

We immediately get the following consequence:

######  Corollary 4.8.

Let $M\in\mathbb{R}_{>0}^{\mathbb{N}}$ and $\Phi\in\mathbb{R}_{\geq
0}^{\mathbb{N}}$ be given such that the requirements of the particular case in
Theorem 4.7 are valid. If $\lim_{j\rightarrow\infty}\frac{\Phi_{j}}{j}=0$,
then $\mathcal{E}_{[M,\Phi]}$ cannot be identified with any weight matrix
class $\mathcal{E}_{[\mathcal{L}]}$ with $\mathcal{L}$ being
$(\mathcal{M}_{\operatorname{sc}})$.

###### Proof.

(i):

Choose $c$ such that $M^{(c,\Phi)}$ is log-convex and normalized. By applying
the optimal functions $\theta_{N}$ from (4.4) to $N\equiv M^{(c,\Phi)}$ we get
by the equality of the classes that there exist $h,x_{0}$ (w.l.o.g. greater
than $1$) such that

(4.5) $\forall x\geq x_{0}~{}\forall j\in\mathbb{N}:\quad M_{j}\leq
c^{\Phi_{j}}M_{j}\leq h^{j+1}L^{(x)}_{j},$

and therefore we get

(4.6) $\forall x\geq x_{0}~{}\forall
j\in\mathbb{N}:\quad\frac{1}{h^{2}}\leq\left(\frac{L^{(x)}_{j}}{M_{j}}\right)^{1/j}.$

In addition we infer, again by working with optimal functions, but now for the
sequences $j\mapsto n^{j}L^{(n)}_{j}$, that for all $n\in\mathbb{N}$ there
exists $c_{n}$ (w.l.o.g. increasing in $n)$ such that

$\forall\;j\in\mathbb{N}:\quad n^{j}L^{(n)}_{j}\leq c_{n}^{\Phi_{j}}M_{j},$

and by taking roots we end up with

(4.7) $\forall\;j\in\mathbb{N}_{>0}:\quad n(L^{(n)}_{j})^{1/j}\leq
c_{n}^{\Phi_{j}/j}M_{j}^{1/j}.$

Now let us assume that (3.1) is violated, i.e., that

$\liminf_{j\rightarrow\infty}\frac{\Phi_{j}}{j}=0,$

then we can find a sequence of integers $j_{n}$ such that

$c_{n}^{\Phi_{j_{n}}/j_{n}}\leq 2.$

Combining (4.6) and (4.7) we infer (for all $n\in\mathbb{N}$ with $n\geq
x_{0}$)

$\frac{n}{h^{2}}\leq
n\left(\frac{L^{(n)}_{j_{n}}}{M_{j_{n}}}\right)^{1/j_{n}}\leq 2,$

which yields a contradiction as $n\rightarrow\infty$.

(ii):

By Proposition 4.5, we infer that the spaces are isomorphic as Fréchet spaces.
Thus we get that for any compact set $K\subset\subset\mathbb{R}$ there exist
$h,x_{0}>0$ and a compact set $J\subset\subset\mathbb{R}$ such that for all
$f\in\mathcal{E}_{(M,\Phi)}(\mathbb{R})=\mathcal{E}_{(\mathcal{L})}(\mathbb{R})$
we have

$\|f\|_{M,\Phi,K,1}\leq\frac{1}{h}\|f\|_{L^{(x_{0})},J,h},$

which yields, by plugging in the family of functions $f_{s}(t):=e^{ist}$,

$\exp(\omega_{M}(s))\leq\frac{1}{h}\exp(\omega_{L^{(x_{0})}}(s/h)).$

Due to log-convexity we can apply (2.2) and get from this estimate (since the
sequences of $\mathcal{L}$ are pointwise ordered), that for all $x\leq x_{0}$
and $j\in\mathbb{N}$

$h^{j+1}L^{(x)}_{j}\leq M_{j}$

and finally, since w.l.o.g. $h\leq 1$, we get that for all $x\leq x_{0}$ and
$j\in\mathbb{N}_{>0}$

(4.8) $h^{2}\leq\left(\frac{M_{j}}{L^{(x)}_{j}}\right)^{1/j}.$

Analogously we argue to get that for all $n\in\mathbb{N}$ there exists
$c_{n}>0$ such that

$c_{n}^{\Phi_{j}+1}M_{j}\leq\left(\frac{1}{n}\right)^{j}L^{(1/n)}_{j},$

and therefore

(4.9)
$\left(\frac{M_{j}}{L_{j}^{(1/n)}}\right)^{1/j}\leq\frac{1}{n}\left(\frac{1}{c_{n}}\right)^{(\Phi_{j}+1)/j}.$

Now again assume that (3.1) is violated, i.e., that

$\liminf_{j\rightarrow\infty}\frac{\Phi_{j}}{j}=0,$

then we can find a sequence of integers $j_{n}$ such that

$\left(\frac{1}{c_{n}}\right)^{(\Phi_{j_{n}}+1)/j_{n}}\leq 2.$

Combining (4.8) and (4.9) we infer

$h^{2}\leq\frac{2}{n},$

which again gives the desired contradiction and thus finishes the proof. ∎

In particular, if we choose for the matrix $\mathcal{L}$ the concrete matrix
$\mathcal{M}_{M,\Phi}$ from (4.1), then we can draw the same conclusion i.e.,
that $\Phi$ already has to be an exponent sequence. Under somewhat milder
conditions, we can actually show even more in this case. This is due to the
fact that we can prove the desired implication directly, however by using the
same techniques ((4.4), Proposition 4.5) as in the proof of Theorem 4.7
before.

######  Theorem 4.9.

Let $M\in\mathbb{R}_{>0}^{\mathbb{N}}$ and $\Phi\in\mathbb{R}_{\geq
0}^{\mathbb{N}}$ be given and $\mathcal{M}_{M,\Phi}$ be the matrix defined in
(4.1). Then we get:

  * $(i)$

The Roumieu case. Assume that $M^{(c,\Phi)}$ is log-convex and normalized for
some $c>0$ and that (as sets)

$\mathcal{E}_{\\{\mathcal{M}_{M,\Phi}\\}}\subseteq\mathcal{E}_{\\{M,\Phi\\}}$

is valid. Then $\Phi$ has to satisfy (3.1). In particular this implication
holds for any $M\in\hyperlink{LCset}{\mathcal{LC}}$.

  * $(ii)$

The Beurling case. Assume that
$\lim_{j\rightarrow+\infty}(M^{(c,\Phi)}_{j})^{1/j}=+\infty$ for all $c>0$ and
such that (w.l.o.g., cf. Remark 4.3) each $M^{(c,\Phi)}$ is log-convex and
$M^{(c,\Phi)}_{0}=1$. Assume also that (as sets and thus automatically as
Fréchet spaces)

$\mathcal{E}_{(M,\Phi)}\subseteq\mathcal{E}_{(\mathcal{M}_{M,\Phi})}$

is valid. Then $\Phi$ has to satisfy (3.1).

Consequently, (3.1) has to hold when assuming (4.3) if the additional
requirements on $M^{(c,\Phi)}$ of the particular case hold true.

If the symmetric restriction from above is imposed on the sequence $\Phi$,
i.e., the growth of $h^{\Phi_{j}}$ is at most geometric, then we recover the
classical ultradifferentiable classes defined by a single weight sequence.

######  Proposition 4.10.

Let $M\in\mathbb{R}_{>0}^{\mathbb{N}}$ and
$\Phi=(\Phi_{j})_{j}\in\mathbb{R}_{\geq 0}^{\mathbb{N}}$ be given. If $\Phi$
is an exponent sequence and in addition also

(4.10) $\limsup_{j\rightarrow\infty}\frac{\Phi_{j}}{j}<\infty,$

then as locally convex vector spaces
$\mathcal{E}_{[M,\Phi]}=\mathcal{E}_{[M]}$.

Finally, we can treat the converse statement.

######  Theorem 4.11.

Let $M\in\mathbb{R}_{>0}^{\mathbb{N}}$ and $\Phi\in\mathbb{R}_{\geq
0}^{\mathbb{N}}$ be given. Then we get:

  * $(i)$

The Roumieu case. Assume that $M^{(c,\Phi)}$ is log-convex and normalized for
some $c>1$, and let $M\in\hyperlink{LCset}{\mathcal{LC}}$ or even $M$ be only
normalized and log-convex. Assume that

$\mathcal{E}_{\\{M,\Phi\\}}(\mathbb{R})=\mathcal{E}_{\\{M\\}}(\mathbb{R})$

is valid (as sets). Then $\Phi$ has to satisfy both (3.1) and (4.10).

  * $(ii)$

The Beurling case. Assume that
$\lim_{j\rightarrow+\infty}(M^{(c,\Phi)}_{j})^{1/j}=+\infty$ for all $c>0$ and
such that (w.l.o.g., cf. Remark 4.3) each $M^{(c,\Phi)}$ is log-convex and
$M^{(c,\Phi)}_{0}=1$. Assume also that (as sets/locally convex vector spaces)

$\mathcal{E}_{(M,\Phi)}(\mathbb{R})=\mathcal{E}_{(M)}(\mathbb{R})$

is valid. Then $\Phi$ has to satisfy both (3.1) and (4.10).

###### Proof.

We follow the proof of Theorem 4.7 with $M=L^{(x)}$ for any $x>0$. In the
Roumieu case the second part in (4.5) implies (4.10) for $\Phi$. Then we
consider $j\mapsto h^{j}M_{j}$ for some arbitrary but fixed $h>1$ instead of
$j\mapsto n^{j}L^{(n)}_{j}$ and so (4.7) yields (3.1).

In the Beurling case note that $M^{(1,\Phi)}=M$ and by the assumption $M$ has
all assumptions from the set $\mathcal{LC}$ except $M_{0}\leq M_{1}$. Then
follow again the proof of Theorem 4.7: Replace $M$ by
$(c^{\Phi_{j}}M_{j})_{j}$ for some $1>c>0$ and $L^{(x)}=M$ and get with $h<1$
in the first step that (4.10) holds. The argument for obtaining (3.1) follows
similarly. ∎

###  4.3. More general (matrix) situations

For the sake of completeness let us comment on even more general situations
compared with the definitions in Section 3. On the one hand, we can consider
$\Phi$-ultradifferentiable classes $\mathcal{E}_{[\mathcal{N},\Phi]}$ defined
in terms of a given weight matrix $\mathcal{N}=\\{N^{(x)}:x>0\\}$, i.e., with
an additional parameter. For $U\subseteq\mathbb{R}^{d}$ non-empty open, these
classes are defined in the natural way by

$\mathcal{E}_{\\{\mathcal{N},\Phi\\}}(U)=\underset{K\subset\subset
U}{\varprojlim}\;\underset{x\in\mathcal{I}}{\varinjlim}\;\underset{h>0}{\varinjlim}\;\mathcal{E}_{N^{(x)},\Phi,h}(K).$

Similarly, we consider for the Beurling case

$\mathcal{E}_{(\mathcal{N},\Phi)}(U)=\underset{K\subset\subset
U}{\varprojlim}\;\underset{x\in\mathcal{I}}{\varprojlim}\;\underset{h>0}{\varprojlim}\;\mathcal{E}_{N^{(x)},\Phi,h}(K).$

Accordingly, we introduce (cf. (4.1))

(4.11) $\mathcal{M}_{\mathcal{N},\Phi}:=\\{N^{(c,c,\Phi)}:c>0\\},\hskip
15.0ptN^{(c,c,\Phi)}_{j}:=c^{\Phi_{j}}N^{(c)}_{j},\;\;\;j\in\mathbb{N}.$

It is then straight-forward to check that Lemma 4.1 can be transferred to this
setting and Theorem 4.4 takes the following form:

######  Theorem 4.12.

Let $\mathcal{N}=\\{N^{(x)}:x>0\\}$ be given and let $\Phi$ be an exponent
sequence. Let $\mathcal{M}_{\mathcal{N},\Phi}$ be the matrix defined in
(4.11), then as locally convex vector spaces we get

(4.12)
$\mathcal{E}_{\\{\mathcal{N},\Phi\\}}=\mathcal{E}_{\\{\mathcal{M}_{\mathcal{N},\Phi}\\}},\hskip
15.0pt\mathcal{E}_{(\mathcal{N},\Phi)}=\mathcal{E}_{(\mathcal{M}_{\mathcal{N},\Phi})}.$

In both cases we can replace the symbol (functor) $\mathcal{E}$ by
$\mathcal{B}$, $\mathcal{D}$, $\mathcal{A}$ or by $\Lambda$.

These classes will be relevant for the study of PTT-limit classes in Section
6. Theorem 4.12 and the matrix introduced in (4.11) should be compared with
the matrix $\mathcal{M}^{\sigma}$, see (6.1); in particular, this result
becomes relevant for the equalities in Remark 6.3.

On the other hand, take $M\in\mathbb{R}_{>0}^{\mathbb{N}}$ and let
$\mathcal{F}:=\\{\Phi^{a}:a>0\\}$ be a family of sequences
$\Phi^{a}\in\mathbb{R}_{\geq 0}^{\mathbb{N}}$ such that

(4.13) $\forall\;0<a\leq
b\;\forall\;j\in\mathbb{N}:\;\;\;\Phi^{a}_{j}\log(a)\leq\Phi^{b}_{j}\log(b).$

We introduce the following locally convex vector spaces

$\mathcal{E}_{\\{M,\mathcal{F}\\}}(K):=\underset{a>0}{\varinjlim}\;\underset{h>0}{\varinjlim}\;\mathcal{E}_{M,\Phi^{a},h}(K),$

and

$\mathcal{E}_{\\{M,\mathcal{F}\\}}(U)=\underset{K\subset\subset
U}{\varprojlim}\;\underset{a>0}{\varinjlim}\;\underset{h>0}{\varinjlim}\;\mathcal{E}_{M,\Phi^{a},h}(K)=\underset{K\subset\subset
U}{\varprojlim}\;\mathcal{E}_{\\{M,\mathcal{F}\\}}(K).$

Similarly, we set

$\mathcal{E}_{(M,\mathcal{F})}(K):=\underset{a>0}{\varprojlim}\;\underset{h>0}{\varprojlim}\;\mathcal{E}_{M,\Phi^{a},h}(K),$

and

$\mathcal{E}_{(M,\mathcal{F})}(U)=\underset{K\subset\subset
U}{\varprojlim}\;\underset{a>0}{\varprojlim}\;\underset{h>0}{\varprojlim}\;\mathcal{E}_{M,\Phi^{a},h}(K)=\underset{K\subset\subset
U}{\varprojlim}\;\mathcal{E}_{(M,\Phi^{a})}(K).$

Finally, let us introduce the matrix

(4.14) $\mathcal{M}_{M,\mathcal{F}}:=\\{M^{(c,\Phi^{c})}:c>0\\},\hskip
25.0ptM^{(c,\Phi^{c})}_{j}:=c^{\Phi^{c}_{j}}M_{j},\;\;\;j\in\mathbb{N}.$

If $\Phi^{a}=\Phi$ for all $a>0$, then (4.13) is trivially satisfied (recall
that $\Phi_{j}\geq 0$) and
$\mathcal{E}_{[M,\mathcal{F}]}=\mathcal{E}_{[M,\Phi]}$ as locally convex
vector spaces.

Theorem 4.4 turns in the following form:

######  Theorem 4.13.

Let $M\in\mathbb{R}_{>0}^{\mathbb{N}}$ be given and let
$\mathcal{F}:=\\{\Phi^{a}:a>0\\}$ be a family of sequences
$\Phi^{a}\in\mathbb{R}_{\geq 0}^{\mathbb{N}}$ satisfying (4.13). Let
$\mathcal{M}_{M,\mathcal{F}}$ be the matrix defined in (4.14) and assume that
this matrix satisfies $(\mathcal{M}_{\\{\operatorname{L}\\}})$ resp.
$(\mathcal{M}_{(\operatorname{L})})$. Then as locally convex vector spaces we
get

(4.15)
$\mathcal{E}_{\\{M,\mathcal{F}\\}}=\mathcal{E}_{\\{\mathcal{M}_{M,\mathcal{F}}\\}},\hskip
20.0pt\mathcal{E}_{(M,\mathcal{F})}=\mathcal{E}_{(\mathcal{M}_{M,\mathcal{F}})}.$

Again, in both cases we can replace the symbol (functor) $\mathcal{E}$ by
$\mathcal{B}$, $\mathcal{D}$, $\mathcal{A}$ or by $\Lambda$.

Proof. Analogous to Theorem 4.4.∎

Let us characterize now the crucial conditions
$(\mathcal{M}_{\\{\operatorname{L}\\}})$ resp.
$(\mathcal{M}_{(\operatorname{L})})$ in terms of a growth condition on
$\mathcal{F}$. The next result generalizes Lemma 4.1 to the matrix
$\mathcal{M}_{M,\mathcal{F}}$ defined in (4.14).

######  Proposition 4.14.

Let $\mathcal{M}_{M,\mathcal{F}}$ be given and assume that
$\mathcal{F}=\\{\Phi^{a}:a>0\\}$ satisfies (4.13).

  * $(a)$

The following are equivalent (Roumieu case):

    * $(i)$

$\mathcal{M}_{M,\mathcal{F}}:=\\{M^{(c,\Phi^{c})}:c>0\\}$ satisfies
$(\mathcal{M}_{\\{\operatorname{L}\\}})$.

    * $(ii)$

The family $\mathcal{F}$ satisfies

(4.16)
$\exists\;\epsilon>0\;\forall\;c>0\;\exists\;d>c:\;\;\;\liminf_{j\rightarrow\infty}\frac{\Phi_{j}^{d}}{j}\log(d)-\frac{\Phi_{j}^{c}}{j}\log(c)\geq\epsilon.$

  * $(b)$

The following are equivalent (Beurling case):

    * $(i)$

$\mathcal{M}_{M,\mathcal{F}}:=\\{M^{(c,\Phi^{c})}:c>0\\}$ satisfies
$(\mathcal{M}_{(\operatorname{L})})$.

    * $(ii)$

The family $\mathcal{F}$ satisfies

(4.17)
$\exists\;\epsilon>0\;\forall\;c>0\;\exists\;d<c:\;\;\;\liminf_{j\rightarrow\infty}\frac{\Phi_{j}^{c}}{j}\log(c)-\frac{\Phi_{j}^{d}}{j}\log(d)\geq\epsilon.$

Proof.

$(a)(i)\Rightarrow(ii)$ By assumption we have (recall
$M^{(c,\Phi^{c})}_{j}:=c^{\Phi^{c}_{j}}M_{j}$):

$\forall\;h>0\;\forall\;c>0\;\exists\;d>0\;\exists\;D\geq
1\;\forall\;j\in\mathbb{N}:\;\;\;h^{j}c^{\Phi^{c}_{j}}M_{j}\leq
Dd^{\Phi^{d}_{j}}M_{j}.$

Fix now $h>1$ and by (4.13) we can assume that $d\geq c$. Hence

$\forall\;c>0\;\exists\;d>c\;\exists\;D\geq
1\;\forall\;j\in\mathbb{N}_{>0}:\;\;\;\log(h)-\frac{\log(D)}{j}\leq\frac{\Phi_{j}^{d}}{j}\log(d)-\frac{\Phi_{j}^{c}}{j}\log(c),$

and so (4.16) is verified with (e.g.) $\epsilon:=\log(h)/2$.

$(a)(ii)\Rightarrow(i)$ (4.16) implies

$\exists\;\epsilon>0\;\forall\;c>0\;\exists\;d>c\;\exists\;j_{c}\in\mathbb{N}\;\forall\;j\geq
j_{c}:\;\;\;\frac{\Phi_{j}^{d}}{j}\log(d)-\frac{\Phi_{j}^{c}}{j}\log(c)\geq\frac{\epsilon}{2}\Leftrightarrow
e^{\epsilon j/2}c^{\Phi^{c}_{j}}M_{j}\leq d^{\Phi^{d}_{j}}M_{j}.$

Then let $h>1$ be given (large) and iterate the previous estimate $n$-times,
with $n\in\mathbb{N}_{>0}$ chosen minimal such that $e^{n\epsilon/2}\geq h$.
This then yields choices $d=c_{n+1}>c_{n}>\dots>c_{1}=c$ (since by assumption
the value of $\epsilon$ is not depending on the choice for $c_{i}$) such that

$h^{j}c^{\Phi^{c}_{j}}M_{j}\leq e^{n\epsilon j/2}c^{\Phi^{c}_{j}}M_{j}\leq
d^{\Phi^{d}_{j}}M_{j}$

for all $j\geq\max\\{j_{c_{i}}:1\leq i\leq n\\}$. Finally, when choosing
$D\geq 1$ sufficiently large, we ensure

$h^{j}c^{\Phi^{c}_{j}}M_{j}\leq Dd^{\Phi^{d}_{j}}M_{j}$

for all $j\in\mathbb{N}$ and $D$ is only depending on the number of iterations
$n$, i.e., on given $h$, and on the given index $c$. Thus
$(\mathcal{M}_{\\{\operatorname{L}\\}})$ is verified.

The equivalence for the Beurling case is analogous. ∎

######  Remark 4.15.

We comment on some special cases:

  * $(a)$

The constant case: If $\Phi^{a}=\Phi$ for all $a>0$ and if $\Phi$ is an
exponent sequence, i.e., (3.1) is valid, then both (4.16) and (4.17) hold
true: For given $c>0$ e.g. we choose $d=2c$ in (4.16) resp. $d=c/2$ in (4.17)
and get both requirements with
$\epsilon:=\log(2)\liminf_{j\rightarrow\infty}\frac{\Phi_{j}}{j}$.

  * $(b)$

Assume that for all $c,d>0$ with $c\leq d$ we have that
$\Phi^{c}\leq\Phi^{d}$, which implies (4.13).

    * $(*)$

Assume that there exists some $c_{0}>0$ such that $\Phi^{c_{0}}$ satisfies
(3.1) with value $\epsilon_{0}>0$. So each $\Phi^{d}$, $d\geq c_{0}$,
satisfies (3.1) with
$\liminf_{j\rightarrow\infty}\frac{\Phi^{d}_{j}}{j}\geq\epsilon_{0}$. Then,
arguing as in the constant case before, we get (4.16) with
$\epsilon:=\log(2)\epsilon_{0}$ for all choices $c\geq c_{0}$. Note that in
the Roumieu case we can omit all $c<c_{0}$ without changing the corresponding
function class.

    * $(*)$

If for all $c>0$ we have that $\Phi^{c}$ satisfies (3.1) uniformly in $c$,
i.e.,

$\exists\;\epsilon_{1}>0\;\forall\;c>0:\;\;\;\liminf_{j\rightarrow\infty}\frac{\Phi^{c}_{j}}{j}\geq\epsilon_{1},$

then (4.17) holds true with $\epsilon:=\log(2)\epsilon_{1}$.

##  5\. PTT-classes as spaces defined by weight matrices

Let the parameters $\tau>0$ and $\sigma>1$ be given but from now on fixed and
consider (with the convention $0^{0}:=1$)

(5.1) $M_{j}=M^{\tau,\sigma}_{j}:=j^{\tau j^{\sigma}},\hskip
15.0pt\Phi_{j}=j^{\sigma}\qquad\text{for all}\quad j\in\mathbb{N}.$

For these particular choices of $M$ and $\Phi$ we write $M^{(c,\tau,\sigma)}$
for $M^{(c,\Phi)}$. Thus the sequences and the matrix introduced in (4.1) have
the form

(5.2) $M^{(c,\tau,\sigma)}_{j}:=M^{(c,\Phi)}_{j}=c^{j^{\sigma}}j^{\tau
j^{\sigma}},\;\;\;c>0,\;j\in\mathbb{N},\qquad\mathcal{M}^{\tau,\sigma}:=\mathcal{M}_{M,\Phi}=\\{M^{(c,\tau,\sigma)}\,\,\,:\;\;\;c>0\\}.$

###  5.1. Properties of the matrix $\mathcal{M}^{\tau,\sigma}$

Note that, in particular, Theorem 4.4 applies to this special situation. We
thus have as a corollary, in accordance with the notation in the works of S.
Pilipović, N. Teofanov, and F. Tomić, the following statement.

######  Proposition 5.1.

Let $U\subseteq\mathbb{R}^{d}$ be open, $\tau>0$, and $\sigma>1$. Then (as
locally convex vector spaces)

(5.3)
$\mathcal{E}_{\\{\tau,\sigma\\}}(U)=\mathcal{E}_{\\{\mathcal{M}^{\tau,\sigma}\\}}(U),\qquad\mathcal{E}_{(\tau,\sigma)}(U)=\mathcal{E}_{(\mathcal{M}^{\tau,\sigma})}(U).$

Therefore, we may apply certain results available in the weight matrix setting
to PTT-classes. First we need to study the properties of the defining weight
matrix $\mathcal{M}^{\tau,\sigma}$.

######  Theorem 5.2.

Let $\tau>0$, and $\sigma>1$ be fixed. Then we have:

  * $(i)$

$\mathcal{M}^{\tau,\sigma}$ satisfies
$(\mathcal{M}_{(\operatorname{C}^{\omega})})$; in fact we even have
$\left(M^{(c,\tau,\sigma)}_{j}/j!^{\alpha}\right)^{1/j}\rightarrow+\infty$ as
$j\rightarrow+\infty$ for any $\alpha>0$ and any $c>0$. Consequently,
$\mathcal{M}^{\tau,\sigma}$ also satisfies $(\mathcal{M}_{\mathcal{H}})$ and
$(\mathcal{M}_{\\{\operatorname{C}^{\omega}\\}})$.

  * $(ii)$

There exists a matrix $\widetilde{\mathcal{M}}^{\tau,\sigma}$ which is
equivalent to $\mathcal{M}^{\tau,\sigma}$ and such that
$\widetilde{\mathcal{M}}^{\tau,\sigma}$ consists only of sequences that are
strongly log-convex (and normalized).

  * $(iii)$

$\mathcal{M}^{\tau,\sigma}$ has both $(\mathcal{M}_{\\{\operatorname{dc}\\}})$
and $(\mathcal{M}_{(\operatorname{dc})})$.

  * $(iv)$

$\mathcal{M}^{\tau,\sigma}$ has both
$(\mathcal{M}_{\\{\operatorname{rai}\\}})$ and
$(\mathcal{M}_{(\operatorname{rai})})$; in fact in both conditions we can
choose the same index $\alpha=\beta$.

  * $(v)$

$\mathcal{M}^{\tau,\sigma}$ has both
$(\mathcal{M}_{\\{\operatorname{FdB}\\}})$ and
$(\mathcal{M}_{(\operatorname{FdB})})$.

  * $(vi)$

For each $c>0$ the sequence $M^{(c,\tau,\sigma)}$ is strongly non-
quasianalytic, in fact we even have that
$\gamma(M^{(c,\tau,\sigma)})=+\infty$. For the precise definition, properties
and meanings of the growth index $\gamma(M)$ introduced in [34, Sect. 1.3] we
refer to [7, Sect. 3].

  * $(vii)$

$\mathcal{M}^{\tau,\sigma}$ neither has
$(\mathcal{M}_{\\{\operatorname{mg}\\}})$ nor
$(\mathcal{M}_{(\operatorname{mg})})$.

  * $(viii)$

The sequences $M^{(c,\tau,\sigma)}$ are pairwise non-equivalent. More
precisely, we have
$M^{(c_{1},\tau,\sigma)}\hyperlink{triangle}{\vartriangleleft}M^{(c_{2},\tau,\sigma)}$
for all $0<c_{1}<c_{2}$.

Proof. $(i)$ For all $j\geq 1$ we see that

$\left(m^{(c,\tau,\sigma)}_{j}\right)^{1/j}=\left(\frac{M^{(c,\tau,\sigma)}_{j}}{j!}\right)^{1/j}\geq\left(\frac{M^{(c,\tau,\sigma)}_{j}}{j^{j}}\right)^{1/j}=\left(c^{j^{\sigma}}j^{\tau
j^{\sigma}-j}\right)^{1/j}=c^{j^{\sigma-1}}j^{\tau j^{\sigma-1}-1},$

and so $\left(m^{(c,\tau,\sigma)}_{j}\right)^{1/j}\rightarrow+\infty$ as
$j\rightarrow\infty$ for all $c>0$. This also implies
$\mu^{(c,\tau,\sigma)}_{j}/j\rightarrow+\infty$ for all $c>0$; see (2.1). More
generally, for any $\alpha>0$ and any $c>0$ it is immediate by the same
estimate above that
$\left(M^{(c,\tau,\sigma)}_{j}/j!^{\alpha}\right)^{1/j}\rightarrow+\infty$.

$(ii)$ Obviously, by the convention $0^{0}:=1$, we get
$1=M^{(c,\tau,\sigma)}_{0}$ and $c=M^{(c,\tau,\sigma)}_{1}$ for all $c>0$.
Thus for each $c\geq 1$ the sequence $M^{(c,\tau,\sigma)}$ is log-convex
because $M$ is log-convex and $j\mapsto j^{\sigma}$ is convex. Actually
$M^{(c,\tau,\sigma)}\in\hyperlink{LCset}{\mathcal{LC}}$ for each $c\geq 1$.

Let us verify that the quotients of each sequence $m^{(c,\tau,\sigma)}$ are
non-decreasing from some index $j_{c}$ on. This then implies the statement,
since each $M^{(c,\tau,\sigma)}$ can then be replaced by an equivalent
sequence $\widetilde{M}^{(c,\tau,\sigma)}$ which is even strongly log-convex
and normalized:

By $(i)$ we have $\mu^{(c,\tau,\sigma)}_{j}/j\rightarrow+\infty$ and so
$\mu^{(c,\tau,\sigma)}_{j}/j\geq 1$ for all $j\geq j^{\prime}_{c}$. Then take
$j^{\prime\prime}_{c}:=\max\\{j_{c},j^{\prime}_{c}\\}$ and set

$\frac{\widetilde{\mu}^{(c,\tau,\sigma)}_{j}}{j}:=1,\;\;\;1\leq
j<j^{\prime\prime}_{c},\hskip
15.0pt\frac{\widetilde{\mu}^{(c,\tau,\sigma)}_{j}}{j}:=\frac{\mu^{(c,\tau,\sigma)}_{j}}{j},\;\;\;j\geq
j^{\prime\prime}_{c}.$

Since $c\mapsto j^{\prime\prime}_{c}$ is non-decreasing (by the order of
$\mu^{(c,\tau,\sigma)}$) we have that $\widetilde{m}^{(c,\tau,\sigma)}$ are
ordered (even w.r.t. their quotient sequences). Moreover,
$M^{(c,\tau,\sigma)}\hyperlink{approx}{\approx}\widetilde{M}^{(c,\tau,\sigma)}$
for each $c>0$ (even on the level of the corresponding quotient sequences).

So let us show that the quotients of $m^{(c,\tau,\sigma)}$ are eventually non-
decreasing for any fixed $c>0$:

$j\mapsto\frac{m^{(c,\tau,\sigma)}_{j}}{m^{(c,\tau,\sigma)}_{j-1}}$ is non-
decreasing if and only if
$j\mapsto\log\left(\frac{m^{(c,\tau,\sigma)}_{j}}{m^{(c,\tau,\sigma)}_{j-1}}\right)$
is so. For all $j\geq 1$ we get

$\frac{m^{(c,\tau,\sigma)}_{j}}{m^{(c,\tau,\sigma)}_{j-1}}=\mu^{(c,\tau,\sigma)}_{j}\frac{1}{j}=c^{j^{\sigma}-(j-1)^{\sigma}}\frac{j^{\tau
j^{\sigma}}}{(j-1)^{\tau(j-1)^{\sigma}}}\frac{1}{j},$

and we set now $f(t):=\log(c)(t^{\sigma}-(t-1)^{\sigma})+\tau
t^{\sigma}\log(t)-\tau(t-1)^{\sigma}\log(t-1)-\log(t)$, $t>1$. Then for all
$t>1$:

$\displaystyle f^{\prime}(t)$
$\displaystyle=\sigma\log(c)(t^{\sigma-1}-(t-1)^{\sigma-1})+\sigma\tau
t^{\sigma-1}\log(t)+\tau t^{\sigma}\frac{1}{t}$
$\displaystyle-\tau\sigma(t-1)^{\sigma-1}\log(t-1)-\tau(t-1)^{\sigma}\frac{1}{t-1}-\frac{1}{t}$
$\displaystyle=(\sigma\log(c)+\tau)(t^{\sigma-1}-(t-1)^{\sigma-1})+\tau\sigma(t^{\sigma-1}\log(t)-(t-1)^{\sigma-1}\log(t-1))-\frac{1}{t}\geq
0$
$\displaystyle\Leftrightarrow(\sigma\log(c)+\tau)t(t^{\sigma-1}-(t-1)^{\sigma-1})+\tau\sigma
t(t^{\sigma-1}\log(t)-(t-1)^{\sigma-1}\log(t-1))\geq 1.$

We now continue to show

(5.4) $(\sigma\log(c)+\tau)t(t^{\sigma-1}-(t-1)^{\sigma-1})+\tau\sigma
t\log(t-1)(t^{\sigma-1}-(t-1)^{\sigma-1})\quad\rightarrow\quad\infty,~{}t\rightarrow\infty,$

which obviously implies the above statement.

Observe that $t^{\sigma-1}-(t-1)^{\sigma-1}=(\sigma-1)\xi^{\sigma-2}$ for some
$\xi\in(t-1,t)$. Therefore, for $t>2$ and $\sigma>2$, we have

$(\sigma-1)t^{\sigma-2}\geq
t^{\sigma-1}-(t-1)^{\sigma-1}\geq(\sigma-1)(t-1)^{\sigma-2},$

whereas in the case $1<\sigma\leq 2$, we have

$(\sigma-1)(t-1)^{\sigma-2}\geq
t^{\sigma-1}-(t-1)^{\sigma-1}\geq(\sigma-1)t^{\sigma-2}.$

Plugging in the appropriate term into (5.4) finishes the proof.

$(iii)$ We claim that for some choices $c,c_{1}>0$ and some $A\geq 1$ (large)
we get for all $j\in\mathbb{N}_{>0}$:

$\displaystyle M^{(c,\tau,\sigma)}_{j+1}\leq
AM^{(c_{1},\tau,\sigma)}_{j}\Leftrightarrow
c^{(j+1)^{\sigma}}(j+1)^{\tau(j+1)^{\sigma}}\leq Ac_{1}^{j^{\sigma}}j^{\tau
j^{\sigma}}$
$\displaystyle\Leftrightarrow(j+1)^{\sigma}\log(c)+\tau(j+1)^{\sigma}\log(j+1)\leq\log(A)+j^{\sigma}\log(c_{1})+\tau
j^{\sigma}\log(j)$
$\displaystyle\Leftrightarrow\tau\underbrace{\frac{(j+1)^{\sigma}\log(j+1)-j^{\sigma}\log(j)}{(j+1)-j}}_{=:\Delta_{j}}\leq\log(A)+j^{\sigma}\log(c_{1})-(j+1)^{\sigma}\log(c).$

Note that the very first estimate is clear for $j=0$ by the convention
$0^{0}:=1$ and when taking $A\geq c^{\sigma}$. Set $f(t):=t^{\sigma}\log(t)$,
$t\geq 1$, and then $f^{\prime}(t)=\sigma
t^{\sigma-1}\log(t)+t^{\sigma-1}=t^{\sigma-1}(\sigma\log(t)+1)$ is strictly
increasing (and tending to infinity as $t\rightarrow+\infty$). Thus we have
$\Delta_{j}\leq f^{\prime}(j+1)=(j+1)^{\sigma-1}(\sigma\log(j+1)+1)\leq
2\sigma(j+1)^{\sigma-1}\log(j+1)$. On the other hand, when given $c\geq 1$ we
choose $c_{1}:=(2c)^{2^{\sigma}}(>c)$, and then for all $j\geq 1$:

$\displaystyle j^{\sigma}\log(c_{1})-(j+1)^{\sigma}\log(c)$
$\displaystyle=(2j)^{\sigma}\log(2c)-(j+1)^{\sigma}\log(c)$
$\displaystyle\geq(j+1)^{\sigma}\log(2c)-(j+1)^{\sigma}\log(c)=(j+1)^{\sigma}\log(2).$

Thus, in the Roumieu case we are able to conclude when choosing $A\geq 1$
sufficiently large.

When given $c_{1}<1$, then we choose $c:=\frac{c_{1}}{2}$ and get for all
$j\geq 1$:

$\displaystyle j^{\sigma}\log(c_{1})-(j+1)^{\sigma}\log(c)$
$\displaystyle=j^{\sigma}\log(c_{1})-(j+1)^{\sigma}\log(c_{1})+(j+1)^{\sigma}\log(2)$
$\displaystyle\geq(j+1)^{\sigma}\log(2),$

since $\log(c_{1})<0$. This proves the Beurling case.

$(iv)$ First, by Stirling’s formula we get for all $c>0$ and $j\geq 1$:

$\displaystyle c^{j^{\sigma-1}}j^{\tau
j^{\sigma-1}-1}=\frac{c^{j^{\sigma-1}}j^{\tau
j^{\sigma-1}}}{j}=\frac{(M^{(c,\tau,\sigma)}_{j})^{1/j}}{j}$
$\displaystyle\leq(m^{(c,\tau,\sigma)}_{j})^{1/j}\leq\frac{e}{j}(M^{(c,\tau,\sigma)}_{j})^{1/j}=ec^{j^{\sigma-1}}j^{\tau
j^{\sigma-1}-1}.$

Thus $(\mathcal{M}_{\\{\operatorname{rai}\\}})$ follows because we have for
all $1\leq c\leq c_{1}$, $A\geq e$ and $1\leq j\leq k$:

$\displaystyle ec^{j^{\sigma-1}}j^{\tau j^{\sigma-1}-1}\leq
Ac_{1}^{k^{\sigma-1}}k^{\tau k^{\sigma-1}-1}.$

Concerning $(\mathcal{M}_{(\operatorname{rai})})$, let $0<c_{1}\leq c<1$,
$A\geq 1$ and $1\leq j\leq k$, then

$\displaystyle ec_{1}^{j^{\sigma-1}}j^{\tau j^{\sigma-1}-1}\leq
Ac^{k^{\sigma-1}}k^{\tau k^{\sigma-1}-1}$ $\displaystyle\Leftrightarrow
j^{\sigma-1}\log(c_{1})-k^{\sigma-1}\log(c)\leq\log(A/e)+\left(\tau
k^{\sigma-1}-1\right)\log(k)-\left(\tau j^{\sigma-1}-1\right)\log(j).$

So the desired estimate follows by choosing $c_{1}=c$ and $A$ large enough.

Recall that each strongly log-convex sequence satisfying $m_{0}=M_{0}=1$ has
the property that $j\mapsto(m_{j})^{1/j}$ is nondecreasing, compare this with
$(ii)$.

$(v)$ This follows by $(i)$, $(iii)$ and $(iv)$; see [24, Lemma 1 $(1)$].

$(vi)$ By repeating the arguments given in $(ii)$ we see that for each
$\alpha>0$ (and any $c>0$) the mapping
$j\mapsto\mu^{(c,\tau,\sigma)}_{j}\frac{1}{j^{\alpha}}$ is eventually non-
decreasing and by $(i)$ one has
$\left(\frac{M^{(c,\tau,\sigma)}_{j}}{j!^{\alpha}}\right)^{1/j}\rightarrow+\infty$
for any $\alpha>0$.

Thus $\gamma(M^{(c,\tau,\sigma)})=+\infty$ follows for all $c>0$ (however this
implication is in general strict; see [10, Prop. 4.4 $(i)\Rightarrow(ii)$])
and so [7, Thm. 3.11] yields the assertion.

$(vii)$ We test conditions $(\mathcal{M}_{\\{\operatorname{mg}\\}})$ and
$(\mathcal{M}_{(\operatorname{mg})})$ for $j=k\geq 1$. So let $A\geq 1$ and
$c,c_{1}>0$ be given, then:

$\displaystyle(M^{(c,\tau,\sigma)}_{2j})^{1/(2j)}\leq
A(M^{(c_{1},\tau,\sigma)}_{j})^{1/j}\Leftrightarrow
c^{(2j)^{\sigma-1}}(2j)^{\tau(2j)^{\sigma-1}}\leq Ac_{1}^{j^{\sigma-1}}j^{\tau
j^{\sigma-1}}$ $\displaystyle\Leftrightarrow
2^{\sigma-1}j^{\sigma-1}\log(c)+\tau
2^{\sigma-1}j^{\sigma-1}\log(2j)\leq\log(A)+j^{\sigma-1}\log(c_{1})+\tau
j^{\sigma-1}\log(j)$ $\displaystyle\Leftrightarrow
2^{\sigma-1}\left(\log(c)+\tau\log(2)\right)-\log(c_{1})\leq\frac{\log(A)}{j^{\sigma-1}}+\log(j)\tau\left(1-2^{\sigma-1}\right).$

As $j\rightarrow\infty$ the first summand on the right-hand side tends to $0$,
whereas the second one tends to $-\infty$ and so does the right-hand side.
This leads to a contradiction for any choice $c,c_{1}>0$.

$(viii)$ For every $0<c_{1}<c_{2}$, we have that

$\lim_{j\to\infty}\left(\frac{M_{j}^{(c_{2},\tau,\sigma)}}{M_{j}^{(c_{1},\tau,\sigma)}}\right)^{1/j}=\lim_{j\to\infty}\left(\frac{c_{2}}{c_{1}}\right)^{j^{\sigma-1}}=\infty.$

∎

###  5.2. Results for PTT-classes

By Theorem 4.4 we know that $\mathcal{E}_{[\tau,\sigma]}(U)$ can be identified
with the matrix class $\mathcal{E}_{[\mathcal{M}^{\tau,\sigma}]}(U)$. From
$(i)$ and $(iii)$ in Theorem 5.2 it follows immediately that
$\mathcal{E}_{[\mathcal{M}^{\tau,\sigma}]}(U)$ contains the real analytic
functions and the restrictions of entire functions and it is closed with
respect to taking derivatives.

By employing results from various works, let us now give a (non-exhaustive)
list of results that hold for those matrix classes due to the regularity
properties listed in Theorem 5.2.

  * $(a)$

Stability properties ([24]): $\mathcal{E}_{[\mathcal{M}^{\tau,\sigma}]}$ is…

    * –

stable under composition,

    * –

stable under solving ODEs,

    * –

stable under inversion,

    * –

inverse-closed.

This follows since by $(i)$, $(iii)$ and $(iv)$ the classes
$\mathcal{E}_{[\mathcal{M}^{\tau,\sigma}]}$ satisfy all necessary properties
such that Theorems 5 and 6 from [24] are applicable.

  * $(b)$

Almost analytic extensions ([6]):
$\mathcal{E}_{[\mathcal{M}^{\tau,\sigma}]}$-regularity of a function can be
characterized by _almost analytic extension_. This means that a function $f$
is in $\mathcal{E}_{[\mathcal{M}^{\tau,\sigma}]}(U)$ if and only if, for any
_quasiconvex_ domain $V$ relatively compact in $U$, $f|_{V}$ can be extended
to a function $F$ on $\mathbb{C}^{d}$ such that $\overline{\partial}F$ tends
to $0$ sufficiently fast near $V$ (measured in terms of
$\mathcal{M}^{\tau,\sigma}$).

This follows since by $(i)$, $(ii)$ and $(iii)$ we have that
$\widetilde{\mathcal{M}}^{\tau,\sigma}$ is a regular weight matrix in the
sense of [6, Def. 2.6].

  * $(c)$

Image of the Borel map ([18], [32], [31], [16], [17]): We have the following
description of the image of the Borel map:

$j^{\infty}_{0}(\mathcal{E}_{[\mathcal{M}^{\tau,\sigma}]})=\Lambda_{[\mathcal{M}^{\tau,\sigma}]}.$

By $(vi)$ we have that each $M^{(c,\tau,\sigma)}$ has $(\gamma_{1})$ and the
rest follows from the results of aforementioned papers.

  * $(d)$

PTT-classes are not “classical ultradifferentiable spaces” ([23], [29]): By
$(vii)$ and $(viii)$, neither
$\mathcal{E}_{\\{\mathcal{M}^{\tau,\sigma}\\}}(U)$ nor
$\mathcal{E}_{(\mathcal{M}^{\tau,\sigma})}(U)$ coincides (as vector spaces)
with $\mathcal{E}_{\\{M\\}}(U)$, $\mathcal{E}_{\\{\omega\\}}(U)$, or,
respectively, with $\mathcal{E}_{(M)}(U)$, $\mathcal{E}_{(\omega)}(U)$ for any
weight sequence $M$ or any weight function $\omega$; see [23, Prop. 4.6] and
[29, Cor. 3.17].

  * $(e)$

Nuclearity ([29]): By $(iii)$ the classes
$\mathcal{E}_{\\{\mathcal{M}^{\tau,\sigma}\\}}(U)$ and
$\mathcal{E}_{(\mathcal{M}^{\tau,\sigma})}(U)$ are nuclear; see [29, Prop.
7.2].

  * $(f)$

Almost harmonic extensions ([5]):
$\mathcal{E}_{[\mathcal{M}^{\tau,\sigma}]}$-regularity of a function can be
characterized by _almost harmonic extension_. A little simplified, this means
that a pair of functions $\phi_{0},\phi_{1}$ is of
$\mathcal{E}_{[\mathcal{M}^{\tau,\sigma}]}$-regularity (on a subset of
$\mathbb{R}^{d})$, if and only if (locally) there exists a function $\Phi$ (on
a subset of $\mathbb{R}^{d+1}$) such that the restriction of $\Phi$ to
$\mathbb{R}^{d}$ coincides with $\phi_{0}$, and the restriction of
$\partial_{y}\Phi$ to $\mathbb{R}^{d}$ coincides with $\phi_{1}$, and
$\Delta\Phi$ tends to $0$ sufficiently fast near $\mathbb{R}^{d}$ (measured in
terms of $\mathcal{M}^{\tau,\sigma}$).

This is due to the fact that Theorems [5, Thm. 3.1, Thm. 3.2, Thm. 4.6] can be
applied to the classes $\mathcal{E}_{[\mathcal{M}^{\tau,\sigma}]}$: For this
note that $[\mathfrak{M}.1]^{*}_{w}$ holds true by $(ii)$ in Theorem 5.2 (each
strongly log-convex sequence even satisfies $(M.1)^{*}$),
$[\mathfrak{M}.2]^{\prime}$ by $(iii)$ and $(NA)$ is precisely
$(\mathcal{M}_{(\operatorname{C}^{\omega})})$ which is valid by $(i)$.
Finally, also in [5] a matrix $\mathcal{M}$ is called non-quasianalytic if
each $M\in\mathcal{M}$ is non-quasianalytic and this is valid, in particular,
by $(vi)$.

##  6\. PTT-limit classes as spaces defined by weight matrices

In [19], for fixed $\sigma\geq 1$, the authors also consider limits with
respect to the parameter $\tau$, i.e., the spaces
$\mathcal{E}_{\infty,\sigma}$ and $\mathcal{E}_{0,\sigma}$, presented in
Subsection 2.7, which are endowed with the natural inductive resp. projective
limit topology.

The main reason to consider these classes is represented by the fact, that
they are stable with respect to so-called _ultradifferential operators._
Observe here that a function $f$ lies in $\mathcal{E}_{\infty,\sigma}(U)$ if
and only if there exists a uniform $\tau$ such that
$f\in\mathcal{E}_{\\{\tau,\sigma\\}}(K)$ for all $K\subset\subset U$ and thus
the limit classes $\mathcal{E}_{\infty,\sigma}(U)$ do not quite fit in the
realm of Roumieu-type classes defined via weight matrices since quantifiers
are exchanged. In the latter case it is allowed that $\tau$ is also depending
on $K$; see (2.3).

In order to comment on this subtle difference, first for $\sigma\geq 1$ let us
from now on consider the matrix (using the notation from (5.2))

(6.1) $\mathcal{M}^{\sigma}:=\\{M^{(\tau,\tau,\sigma)}:~{}\tau>0\\},\hskip
15.0ptM^{(\tau,\tau,\sigma)}_{j}:=\tau^{j^{\sigma}}j^{\tau j^{\sigma}}.$

We get the following connection with the respective matrix class defined in
terms of $\mathcal{M}^{\sigma}$.

######  Theorem 6.1.

Let $U,V\subseteq\mathbb{R}^{d}$ be open, and $\overline{V}\subset\subset U$.
Then as locally convex vector spaces we get

(6.2)
$\mathcal{E}_{\infty,\sigma}(U)\hookrightarrow\mathcal{E}_{\\{\mathcal{M}^{\sigma}\\}}(U)\hookrightarrow\mathcal{E}_{\infty,\sigma}(V),\hskip
15.0pt\mathcal{E}_{0,\sigma}(U)=\mathcal{E}_{(\mathcal{M}^{\sigma})}(U).$

Proof. The Roumieu case. The first inclusion is clear from the definition of
the respective spaces. For the second one, observe that any
$f\in\mathcal{E}_{\\{\mathcal{M}^{\sigma}\\}}(U)$ lies in
$\mathcal{E}_{\\{M^{(\tau,\tau,\sigma)}\\}}(\overline{V})$ for some $\tau$,
i.e., there exists $h\geq 1$ and $A>0$ such that for all $x\in\overline{V}$
and $\alpha\in\mathbb{N}^{d}$ we have

$|f^{(\alpha)}(x)|\leq Ah^{|\alpha|}M^{(\tau,\tau,\sigma)}_{|\alpha|},$

and for any $\tau^{\prime}>\tau$, we find $B>0$ such that for all $j$ we have

$h^{j}M^{(\tau,\tau,\sigma)}_{j}\leq
BM^{(\tau^{\prime},\tau^{\prime},\sigma)}_{j},$

which finishes the Roumieu case.

The Beurling case. Since there is a universal quantifier for the compact set,
the index $\tau$, and the geometric factor $h$, we do not have to worry about
interchanging those. The rest follows from Proposition 5.1.

∎

######  Proposition 6.2.

Let $\sigma\geq 1$ and $\mathcal{N}=\\{N^{(x)}:x>0\\}$ be a
$(\mathcal{M}_{\operatorname{sc}})$ and non-quasianalytic weight matrix.
Suppose that we have

$\mathcal{E}_{\infty,\sigma}(U)\subseteq\mathcal{E}_{\\{\mathcal{N}\\}}(U).$

Then it follows that

$\mathcal{E}_{\infty,\sigma}(U)\subsetneq\mathcal{E}_{\\{\mathcal{N}\\}}(U).$

###### Proof.

Similarly as in $(viii)$ in Theorem 5.2 we get

$\forall\;\tau_{1}>\tau>0:\;\;\;M^{(\tau,\tau,\sigma)}\lhd
M^{(\tau_{1},\tau_{1},\sigma)}.$

The inclusion
$\mathcal{E}_{\infty,\sigma}(U)\subseteq\mathcal{E}_{\\{\mathcal{N}\\}}(U)$
and the optimal function $\theta_{M^{(\tau_{1},\tau_{1},\sigma)}}$ (see (4.4))
implies the following:

(6.3) $\forall\;\tau>0\;\exists\;x>0:\;\;\;M^{(\tau,\tau,\sigma)}\lhd
N^{(x)}.$

Therefore, note that
$\theta_{M^{(\tau_{1},\tau_{1},\sigma)}}\in\mathcal{E}_{\infty,\sigma}(U)$ for
any $\tau_{1}>0$ since the estimate
$|\theta_{M^{(\tau_{1},\tau_{1},\sigma)}}(t)|\leq
2^{j+1}\tau_{1}^{j^{\sigma}}j^{\tau_{1}j^{\sigma}}$ holds globally on whole
$\mathbb{R}$; see again [35, Thm. 1], [23, Lemma 2.9] and the detailed proof
in [26, Prop. 3.1.2].

Let $K_{j}$ be a sequence of mutually disjoint compact sets with non-empty
interior contained in $U$ such that they accumulate at the boundary of $U$,
i.e., for any compact set $K\subset\subset U$ there exists $j$ such that
$K_{j}\cap K=\emptyset$. Let $S_{j}$ be also a sequence of compact sets such
that $S_{j}\subseteq K_{j}^{\circ}$. Finally, let $x_{j}\in S_{j}$. Then by
[22, Cor. 3.12] there exists
$\phi_{j}\in\mathcal{D}_{\\{N^{(j)}\\}}(K_{j}^{\circ})$ such that
$\phi_{j}\equiv 1$ on $S_{j}$. Let
$\theta_{j}\in\mathcal{E}_{\\{N^{(j)}\\}}(\mathbb{R})$ be such that
$|\theta_{j}^{(k)}((x_{j})_{1})|\geq N^{(j)}_{k}$ (where $(x_{j})_{1}$ is the
first component of $x_{j}$), and set
$\Theta_{j}(x_{1},\dots,x_{d}):=\theta_{j}(x_{1})$. Finally, set

$h_{j}:=\Theta_{j}\phi_{j},\quad h:=\sum_{j}h_{j}.$

Then clearly $h_{j}\in\mathcal{D}_{\\{N^{(j)}\\}}(K_{j}^{\circ})$, and thus
$h\in\mathcal{E}_{\\{\mathcal{N}\\}}(U)$. But
$h\notin\mathcal{E}_{\\{\tau,\sigma\\}}$ for any $\tau$ (and therefore not in
$\mathcal{E}_{\infty,\sigma}(U)$). To see this, take for given
$M^{(\tau,\tau,\sigma)}$ some $j$ big enough to get (6.3). By taking
$K=K_{j}$, one immediately gets $h\notin\mathcal{E}_{\\{\tau,\sigma\\}}(U)$. ∎

######  Remark 6.3.

After a private communication, in the very recent paper [33] the authors
already have taken into account this fact and included the definition of the
limit classes

$\mathcal{E}^{R}_{\infty,\sigma}(U):=\bigcap_{K\subset\subset
U}\bigcup_{\tau>0}\mathcal{E}_{\\{\tau,\sigma\\}}(K)=\bigcap_{K\subset\subset
U}\bigcup_{\tau>0}\mathcal{E}_{\\{\mathcal{M}^{\tau,\sigma}\\}}(K)=\mathcal{E}_{\\{\mathcal{M}^{\sigma}\\}}(U),$

see [33, $(2.12)$]. Note that this difference might be considered negligible
in the light of Theorem 6.1 and for the Beurling-type both notions coincide;
i.e.

$\mathcal{E}_{0,\sigma}(U)=\mathcal{E}_{(\mathcal{M}^{\sigma})}(U).$

When considering the notion of germs of
$\mathcal{E}_{\infty,\sigma}$-functions then also no difference occurs. For
these equalities recall Theorem 4.12.

###  6.1. Properties of the matrix $\mathcal{M}^{\sigma}$

From now on we focus on $\mathcal{E}_{[\mathcal{M}^{\sigma}]}(U)$ and gather
several important growth and regularity properties for the crucial weight
matrix $\mathcal{M}^{\sigma}$ from (6.1).

######  Theorem 6.4.

Let $\sigma>1$, then the matrix $\mathcal{M}^{\sigma}$ has the following
properties:

  * $(i)$

$\mathcal{M}^{\sigma}$ satisfies $(\mathcal{M}_{(\operatorname{C}^{\omega})})$
and, more generally, even
$\left(\frac{M^{(\tau,\tau,\sigma)}_{j}}{j!^{\alpha}}\right)^{1/j}\rightarrow+\infty$
for any $\alpha>0$ and any $\tau>0$.

  * $(ii)$

$\mathcal{M}^{\sigma}$ is equivalent to a matrix
$\widetilde{\mathcal{M}}^{\sigma}$ all of whose sequences are strongly log-
convex.

  * $(iii)$

$\mathcal{M}^{\sigma}$ satisfies both
$(\mathcal{M}_{\\{\operatorname{mg}\\}})$ and
$(\mathcal{M}_{(\operatorname{mg})})$.

  * $(iv)$

$\mathcal{M}^{\sigma}$ has both $(\mathcal{M}_{\\{\operatorname{rai}\\}})$ and
$(\mathcal{M}_{(\operatorname{rai})})$; in fact we have that in both
conditions we can choose the same index.

  * $(v)$

$\mathcal{M}^{\sigma}$ has both $(\mathcal{M}_{\\{\operatorname{FdB}\\}})$ and
$(\mathcal{M}_{(\operatorname{FdB})})$.

  * $(vi)$

$\mathcal{M}^{\sigma}$ has both $(\mathcal{M}_{\\{\operatorname{L}\\}})$ and
$(\mathcal{M}_{(\operatorname{L})})$.

  * $(vii)$

The sequences $M^{(c,c,\sigma)}$ are pair-wise not equivalent and
$\mathcal{M}^{\sigma}$ has both $(\mathcal{M}_{\\{\operatorname{BR}\\}})$ and
$(\mathcal{M}_{(\operatorname{BR})})$ in [23, Sect. 4.1].

  * $(viii)$

Each $M^{(\tau,\tau,\sigma)}$ is strongly non-quasianalytic; in fact we even
have that $\gamma(M^{(\tau,\tau,\sigma)})=+\infty$ for all $\tau>0$.

Proof. In order to shorten the notation, we write in this proof
$M^{(c)}:=M^{(c,c,\sigma)}$ .

$(i)$ This follows just as in the proof of Theorem 5.2.

$(ii)$ By the convention $0^{0}:=1$ we get $1=M^{(c)}_{0}$ and $c=M^{(c)}_{1}$
for all $c>0$. For each $c\geq 1$ the sequence $M^{(c)}$ is log-convex because
$j\mapsto\log(j^{cj^{\sigma}})=cj^{\sigma}\log(j)$ is convex; more precisely
one has $M^{(c)}\in\hyperlink{LCset}{\mathcal{LC}}$ for all $c\geq 1$.

By replacing $\tau$ by $c$, we can repeat the arguments given in the proof of
$(ii)$ in Theorem 5.2. Since also the order is preserved we have that
$\widetilde{\mathcal{M}}^{\sigma}$ is standard log-convex.

$(iii)$ First, we test conditions $(\mathcal{M}_{\\{\operatorname{mg}\\}})$
and $(\mathcal{M}_{(\operatorname{mg})})$ on the diagonal, i.e., for $j=k\geq
1$. So let $A\geq 1$ and $c,c_{1}>0$ be given, then:

$\displaystyle(M^{(c)}_{2j})^{1/(2j)}\leq
A(M^{(c_{1})}_{j})^{1/j}\Leftrightarrow
c^{(2j)^{\sigma-1}}(2j)^{c(2j)^{\sigma-1}}\leq
Ac_{1}^{j^{\sigma-1}}j^{c_{1}j^{\sigma-1}}$
$\displaystyle\Leftrightarrow(2j)^{\sigma-1}\log(c)+c(2j)^{\sigma-1}\log(2j)\leq\log(A)+j^{\sigma-1}\log(c_{1})+c_{1}j^{\sigma-1}\log(j).$

We also have

$\displaystyle c(2j)^{\sigma-1}\log(2j)\leq j^{\sigma-1}c_{1}\log(j)$
$\displaystyle\Leftrightarrow 2^{\sigma-1}\log(2j)\leq\frac{c_{1}}{c}\log(j)$
$\displaystyle\Leftrightarrow
2^{\sigma-1}\log(2)\leq\log(j)(\frac{c_{1}}{c}-2^{\sigma-1}).$

In the Roumieu case, w.l.o.g. we take $c$ large enough to guarantee
$c^{2^{\sigma-1}-1}\geq 1+2^{\sigma-1}$. Thus, when given such an index $c$ we
choose $c_{1}:=c^{2^{\sigma-1}}>c$. Then, on the one hand clearly
$(2j)^{\sigma-1}\log(c)=j^{\sigma-1}\log(c_{1})$ and, on the other hand
$\frac{c_{1}}{c}-2^{\sigma-1}\geq 1\Leftrightarrow c_{1}\geq(1+2^{\sigma-1})c$
because $c_{1}=c^{2^{\sigma-1}}\geq(1+2^{\sigma-1})c\Leftrightarrow
c^{2^{\sigma-1}-1}\geq 1+2^{\sigma-1}$. Thus we are done when choosing $A$
sufficiently large and note that $c_{1}\rightarrow\infty$ as
$c\rightarrow\infty$.

In the Beurling case, w.l.o.g. we take given $c_{1}<1$ small enough to ensure
$c_{1}<\frac{1}{1+2^{\sigma-1}}$ and then we set $c:=c_{1}^{2}<c_{1}$. Then,
on the one hand, $c_{1}\geq c(1+2^{\sigma-1})$ is immediate and, second, we
have (note that $\log(c_{1})<0$)

$(2j)^{\sigma-1}\log(c)\leq j^{\sigma-1}\log(c_{1})\Leftrightarrow
2(2j)^{\sigma-1}\log(c_{1})\leq j^{\sigma-1}\log(c_{1})\Leftrightarrow
2(2j)^{\sigma-1}\geq j^{\sigma-1},$

which holds for all $j\in\mathbb{N}$.

So far we have verified the desired properties on the diagonal (i.e., $j=k$).
However, by the equivalence stated in $(ii)$ before also
$\widetilde{\mathcal{M}}^{\sigma}$ has both
$(\mathcal{M}_{\\{\operatorname{mg}\\}})$ and
$(\mathcal{M}_{(\operatorname{mg})})$ verified on the diagonal. Thus [27, Thm.
9.5.1, Thm. 9.5.3] applied to $\widetilde{\mathcal{M}}^{\sigma}$ yields the
conclusion and by the equivalence we are done with $\mathcal{M}^{\sigma}$,
too.

$(iv)$ $(\mathcal{M}_{\\{\operatorname{rai}\\}})$ follows by repeating the
estimates from $(iv)$ in Theorem 5.2. Note that we have for all $1\leq c\leq
c_{1}$, $A\geq e$ and $1\leq j\leq k$

$\displaystyle ec^{j^{\sigma-1}}j^{cj^{\sigma-1}-1}\leq
Ac_{1}^{k^{\sigma-1}}k^{c_{1}k^{\sigma-1}-1}.$

Let $0<c_{1}\leq c<1$, $A\geq 1$ and $1\leq j\leq k$, then

$\displaystyle ec_{1}^{j^{\sigma-1}}j^{c_{1}j^{\sigma-1}-1}\leq
Ac^{k^{\sigma-1}}k^{ck^{\sigma-1}-1}$ $\displaystyle\Leftrightarrow
j^{\sigma-1}\log(c_{1})-k^{\sigma-1}\log(c)\leq\log(A/e)+\left(ck^{\sigma-1}-1\right)\log(k)-\left(c_{1}j^{\sigma-1}-1\right)\log(j).$

We take $c_{1}=c$ and repeat the computation from $(iv)$ in Theorem 5.2 when
$\tau$ is replaced by $c$. This should be compared with $(ii)$ and recall that
each strongly log-convex sequence satisfying $m_{0}=M_{0}=1$ has the property
that $j\mapsto(m_{j})^{1/j}$ is nondecreasing.

$(v)$ This follows by $(i)$, $(iii)$ and $(iv)$; see [24, Lemma 1 $(1)$].

$(vi)$ For all $h\geq 1$ (large) and all $0<c<c_{1}$ we can find some constant
$A\geq 1$ (large) such that for all $j\in\mathbb{N}_{>0}$:

$h^{j}M^{(c)}_{j}=h^{j}c^{j^{\sigma}}j^{cj^{\sigma}}\leq
Ac_{1}^{j^{\sigma}}j^{c_{1}j^{\sigma}}=AM^{(c_{1})}_{j}\Leftrightarrow h\leq
A^{1/j}\left(\frac{c_{1}}{c}\right)^{j^{\sigma}}j^{j^{\sigma}(c_{1}-c)}.$

$(vii)$ The same estimate as given in $(vi)$ also yields the following
property for $\mathcal{M}^{\sigma}$:

$\forall\;0<c<c_{1}:\;\;\;M^{(c)}\hyperlink{triangle}{\vartriangleleft}M^{(c_{1})},$

hence both desired properties.

$(viii)$ Follows analogously as in $(vi)$ in Theorem 5.2.

∎

###  6.2. PTT-limit classes as Braun-Meise-Taylor classes

Let $\sigma>1$ be given. Then, on the one hand
$\mathcal{E}_{[\mathcal{M}^{\sigma}]}$ cannot be described by a single weight
sequence which follows by $(vii)$ in Theorem 6.4. On the other hand, the aim
of this section is to show that it actually can be understood as a Braun-
Meise-Taylor class. This question has very recently been studied and solved in
[33] (for the modified defined limit classes mentioned in Remark 6.3). There
the authors give precise asymptotics of $\omega$ in terms of the so-called
_Lambert function_ $W$; cf. [33, Prop. 3.1]. However, we give an independent
proof of their main result by involving only weight matrix techniques.

Let us emphasize that for $\sigma=1$ this statement is not true: By (6.1) the
matrix $\mathcal{M}^{1}:=\\{M^{(\tau,\tau,1)}:~{}\tau>0\\}$ consists of
sequences $M^{(\tau,\tau,1)}_{j}=\tau^{j}j^{\tau j}$ and hence
$\mathcal{M}^{1}$ is equivalent to the Gevrey matrix
$\mathcal{G}_{0}=\\{(j!^{\tau})_{j\in\mathbb{N}}:\tau>0\\}$. From (the first
paragraph in the proof of) [23, Thm. 5.22] it follows that the corresponding
weight matrix class cannot be described by a space given by a log-convex $M$
(in particular $M\in\hyperlink{LCset}{\mathcal{LC}}$) or by a weight function
$\omega$.

We prove now an abstract result on the connection between weight sequences and
their associated weight functions which is important in the
ultradifferentiable setting on its own.

######  Lemma 6.5.

Let $M,N\in\hyperlink{LCset}{\mathcal{LC}}$. Then the following are
equivalent:

  * $(i)$

$M$ and $N$ are related by

(6.4) $\exists\;c\in\mathbb{N}_{>0}\;\exists\;A\geq
1\;\forall\;j\in\mathbb{N}:\;\;\;N_{j}\leq A(M_{cj})^{1/c}.$

  * $(ii)$

The associated weight functions are related by

$\omega_{M}(t)=O(\omega_{N}(t)),\;\;\;t\rightarrow+\infty.$

Moreover, the following are equivalent:

  * $(i)^{\prime}$

$M$ and $N$ are related by

(6.5) $\forall\;c\in\mathbb{N}_{>0}\;\exists\;A\geq
1\;\forall\;j\in\mathbb{N}:\;\;\;(N_{cj})^{1/c}\leq AM_{j}.$

  * $(ii)^{\prime}$

The associated weight functions are related by

$\omega_{M}(t)=o(\omega_{N}(t)),\;\;\;t\rightarrow+\infty.$

Proof. $(ii)\Rightarrow(i)$ We have $\omega_{M}(t)\leq c\omega_{N}(t)+c$ for
some $c\geq 1$ (large) and all $t\geq 0$. W.l.o.g. take $c\in\mathbb{N}_{>0}$
and then (2.2) yields for all $j\in\mathbb{N}$:

$\displaystyle M_{cj}$ $\displaystyle=\sup_{t\geq
0}\frac{t^{cj}}{\exp(\omega_{M}(t))}\geq\frac{1}{e^{c}}\sup_{t\geq
0}\frac{t^{cj}}{\exp(c\omega_{N}(t))}=\frac{1}{e^{c}}\left(\sup_{t\geq
0}\frac{t^{j}}{\exp(\omega_{N}(t))}\right)^{c}=\frac{1}{e^{c}}N_{j}^{c}.$

Thus (6.4) is shown with $A=e$.

$(i)\Rightarrow(ii)$ For given $M\in\mathbb{R}_{>0}^{\mathbb{N}}$ and
$c\in\mathbb{N}_{>0}$ we set

$\widetilde{M}^{c}_{j}:=(M_{cj})^{1/c},$

hence $\widetilde{M}^{1}\equiv M$ is clear. If $M$ is log-convex, then each
$\widetilde{M}^{c}$ as well and $\widetilde{M}^{c}_{0}=1$ if $M_{0}=1$. If
$M\in\hyperlink{LCset}{\mathcal{LC}}$, then
$\widetilde{M}^{c}\in\hyperlink{LCset}{\mathcal{LC}}$ (for some/any
$c\in\mathbb{N}_{>0}$). Thus by definition and assumption we get

(6.6) $\exists\;A\geq 1\;\forall\;t\geq
0:\;\;\;\omega_{\widetilde{M}^{c}}(t)\leq\omega_{N}(t)+\log(A).$

We obtain for all $j\in\mathbb{N}$ and $c\in\mathbb{N}_{>0}$

$\widetilde{M}^{c}_{j}:=(M_{cj})^{1/c}=\left(\sup_{t\geq
0}\frac{t^{cj}}{\exp(\omega_{M}(t))}\right)^{1/c}.$

Moreover, since $M\in\hyperlink{LCset}{\mathcal{LC}}$ we get $\omega_{M}(t)=0$
for $t\in[0,1]$ (i.e., normalization) and so:

$\displaystyle\widetilde{M}^{c}_{j}$ $\displaystyle=\sup_{t\geq
0}\frac{t^{j}}{\exp(c^{-1}\omega_{M}(t))}=\sup_{t\geq
1}\frac{t^{j}}{\exp(c^{-1}\omega_{M}(t))}=\exp(\sup_{t\geq
1}j\log(t)-c^{-1}\omega_{M}(t))$ $\displaystyle=\exp(\sup_{s\geq
0}js-c^{-1}\omega_{M}(e^{s}))=:\exp(\varphi^{*}_{c^{-1}\omega_{M}}(j)).$

Thus we may apply [8, Lemma 2.5] to $\omega\equiv c^{-1}\omega_{M}$ and get

$c^{-1}\omega_{M}\hyperlink{sim}{\sim}\omega_{\widetilde{M}^{c}}.$

More precisely, by setting the weight matrix parameter $x=1$, we see

(6.7) $\exists\;D>0\;\forall\;t\geq 0:\;\;\;\omega_{\widetilde{M}^{c}}(t)\leq
c^{-1}\omega_{M}(t)\leq 2\omega_{\widetilde{M}^{c}}(t)+D.$

Combining (6.7) with (6.6) immediately yields

$\frac{1}{2c}\omega_{M}(t)-\frac{D}{2}\leq\omega_{\widetilde{M}^{c}}(t)\leq\omega_{N}(t)+\log(A)\Longrightarrow\omega_{M}(t)\leq
2c\omega_{N}(t)+Dc+2c\log(A).$

Thus $\omega_{M}(t)=O(\omega_{N}(t))$ as $t\rightarrow+\infty$ is verified.

$(ii)^{\prime}\Rightarrow(i)^{\prime}$ For all $c\in\mathbb{N}_{>0}$ we can
find $D\geq 1$ such that $\omega_{M}(t)\leq\frac{1}{c}\omega_{N}(t)+D$ for all
$t\geq 0$ and so, analogously as before, we obtain (6.5) with $A:=e^{D}$.

$(i)^{\prime}\Rightarrow(ii)^{\prime}$ Using the notation from above, (6.6)
transfers into

(6.8) $\forall\;c\in\mathbb{N}_{>0}\;\exists\;A\geq 1\;\forall\;t\geq
0:\;\;\;\omega_{M}(t)\leq\omega_{\widetilde{N}^{c}}(t)+\log(A).$

Then we follow the arguments in $(i)\Rightarrow(ii)$ and combine (6.8) with
the first half from (6.7) applied to $N$ in order to get

$\forall\;c\in\mathbb{N}_{>0}\;\exists\;A\geq 1\;\forall\;t\geq
0:\;\;\;\omega_{M}(t)\leq c^{-1}\omega_{N}(t)+\log(A).$

Thus $\omega_{M}(t)=o(\omega_{N}(t))$ is verified. ∎

The importance of Lemma 6.5 is that it enables the possibility to express all
requirements in [29, Cor. 3.17 $(ii)$] purely in terms of the given matrix
$\mathcal{N}$ directly:

######  Corollary 6.6.

Let $\mathcal{M}$ be $(\mathcal{M}_{\operatorname{sc}})$. Then as locally
convex vector spaces

$\mathcal{E}_{[\mathcal{M}]}=\mathcal{E}_{[\omega]},$

with $\omega$ being a weight function in the sense of Braun-Meise-Taylor (see
[3], [29]) if and only if there exists a $(\mathcal{M}_{\operatorname{sc}})$
matrix $\mathcal{N}=\\{N^{(\alpha)}:\alpha>0\\}$ which is $R$\- resp.
$B$-equivalent to $\mathcal{M}$ and such that

  * $(*)$

$\mathcal{N}$ has $(\mathcal{M}_{[\operatorname{L}]})$,

  * $(*)$

$\mathcal{N}$ has $(\mathcal{M}_{[\operatorname{mg}]})$,

  * $(*)$

$\mathcal{N}$ has (cf. (6.4))

(6.9) $\forall\;\alpha,\beta>0\;\exists\;c\in\mathbb{N}_{>0}\;\exists\;A\geq
1\;\forall\;j\in\mathbb{N}:\;\;\;N^{(\alpha)}_{j}\leq
A(N^{(\beta)}_{cj})^{1/c}.$

######  Remark 6.7.

Any $\omega_{N^{(\alpha)}}$ (for $N^{(\alpha)}\in\mathcal{N}$) is a valid
choice for $\omega$ in Corollary 6.6.

Note that (6.9) is clearly preserved under R- and B-equivalence of weight
matrices.

In particular we can apply this statement to PTT-limit classes and get the
following.

######  Theorem 6.8.

Let $\sigma>1$ and put $\omega^{(\sigma)}:=\omega_{M^{(1,1,\sigma)}}$; i.e.
the associated weight function of
$M^{(1,1,\sigma)}=(j^{j^{\sigma}})_{j\in\mathbb{N}}$. Then as locally convex
vector spaces we get

(6.10)
$\mathcal{E}_{[\mathcal{M}^{\sigma}]}=\mathcal{E}_{[\omega^{(\sigma)}]}.$

Moreover, all associated weight functions of the matrix $\mathcal{M}^{\sigma}$
are equivalent, i.e.,

$\forall\;h,h^{\prime},\tau,\tau^{\prime}>0:\;\;\;\omega_{M^{(h,\tau,\sigma)}}\hyperlink{sim}{\sim}\omega_{M^{(h^{\prime},\tau^{\prime},\sigma)}},$

and consequently in (6.10) we can replace $\omega^{(\sigma)}$ by any
$\omega_{M^{(h,\tau,\sigma)}}$.

###### Proof.

We only have to verify (6.9) in Corollary 6.6 for the matrix
$\mathcal{M}^{\sigma}$ (the first two conditions are contained in Theorem
6.4). But this is clear for our concrete matrix $\mathcal{M}^{\sigma}$ since
(6.9) holds for $\beta\geq\alpha$ taking $c=A=1$ and for $\beta<\alpha$ we
have that

$j\mapsto j^{\sigma}\big{(}\log(\alpha)-c^{\sigma-1}\log(\beta)+(\alpha-\beta
c^{\sigma-1})\log(j)-(\beta c^{\sigma-1})\log(c)\big{)}$

is bounded from above for $c$ large enough. ∎

######  Remark 6.9.

The previous result fails for $\sigma=1$: The Gevrey matrix
$\mathcal{G}_{0}(=\mathcal{M}^{1})$ clearly satisfies
$(\mathcal{M}_{[\operatorname{L}]})$ and $(\mathcal{M}_{[\operatorname{mg}]})$
but (6.9) is violated when taking e.g. $\alpha=2\beta$.

###  6.3. Results for PTT-limit classes

As seen in the previous section the PTT-limit classes can be represented by
certain Braun-Meise-Taylor classes (defined by the weight function
$\omega^{(\sigma)}:=\omega_{M^{(1,1,\sigma)}}$). Let us now give additional
properties available for PTT-limit classes that follow from this
representation (and the properties listed in Theorem 6.4).

  * $(a)$

Stability properties ([24]): $\mathcal{E}_{[\mathcal{M}^{\sigma}]}$ is…

    * –

stable under composition,

    * –

stable under solving ODEs,

    * –

stable under inversion,

    * –

inverse-closed.

This follows since by $(i)$, $(iii)$ and $(iv)$ in Theorem 6.4 the classes
$\mathcal{E}_{[\mathcal{M}^{\sigma}]}$ satisfy all necessary properties such
that Theorems 5 and 6 from [24] are applicable.

  * $(b)$

Almost analytic extensions ([6]):
$\mathcal{E}_{[\mathcal{M}^{\sigma}]}$-regularity of a function can be
characterized by _almost analytic extension_. This means mutatis mutandis the
same as already outlined in Section 5.2.

In addition both [6, $(7.1),(7.2)$] hold true; $(7.1)$ is precisely
$(\mathcal{M}_{\\{\operatorname{mg}\\}})$ and $(7.2)$ is
$(\mathcal{M}_{\\{\operatorname{FdB}\\}})$. In particular one can deduce
(among other things) an ultradifferentiable elliptic regularity theorem.

  * $(c)$

The image of the Borel map ([18], [32], [31], [16], [17]): We have the
following description of the image of the Borel map:

$j^{\infty}_{0}(\mathcal{E}_{[\mathcal{M}^{\sigma}]})=\Lambda_{[\mathcal{M}^{\sigma}]}.$

By $(viii)$ we have that each $M^{(\tau,\tau,\sigma)}$ has $(\gamma_{1})$ and
the rest follows from the results of the aforementioned papers.

  * $(d)$

Cartesian closedness ([28]): For $E_{1},E_{2},F$ _convenient_ vector spaces
and $U_{i}\subseteq E_{i}$ $c^{\infty}$-open (for the definitions consult
[28], or the thorough treatment [12]) one has as convenient vector spaces

$\mathcal{E}_{[\mathcal{M}^{\sigma}]}(U_{1}\times
U_{2},F)\cong\mathcal{E}_{[\mathcal{M}^{\sigma}]}(U_{1},\mathcal{E}_{[\mathcal{M}^{\sigma}]}(U_{2},F)).$

This follow since by $(i)$, $(ii)$, $(iii)$ and $(iv)$ the classes
$\mathcal{E}_{[\mathcal{M}^{\sigma}]}$ form cartesian closed categories due to
[28, Thm. 5.9, 6.2].

  * $(e)$

A result on powers ([14], [15]): If for two integers $p,q$ with $\gcd(p,q)=1$
and some function $f$, we have
$f^{p},f^{q}\in\mathcal{E}_{[\mathcal{M}^{\sigma}]}$, then we already have
$f\in\mathcal{E}_{[\mathcal{M}^{\sigma}]}$.

This follows since by $(iii)$ we have that $\mathcal{M}^{\sigma}$ satisfies
$(\mathcal{M}_{[\operatorname{mg}]})$, and, by $(ii)$,
$\widetilde{\mathcal{M}}^{\sigma}$ has the desired properties. Thus [15, Thm.
1.1, Thm. 4.1] (cf. also [14, Thm. 4.2]) is applicable and immediately gives
the claim.

  * $(f)$

Almost harmonic extensions ([5]):
$\mathcal{E}_{[\mathcal{M}^{\sigma}]}$-regularity can be characterized by
_almost harmonic extension_. The rest is mutatis mutandis the same as outlined
in Section 5.2.

  * $(g)$

Nuclearity ([29]): By $(iii)$ the classes
$\mathcal{E}_{\\{\mathcal{M}^{\sigma}\\}}(U)$ and
$\mathcal{E}_{(\mathcal{M}^{\sigma})}(U)$ are nuclear; see [29, Prop. 7.2].

###  6.4. A further possible result

In [4], the authors consider ultradistributional boundary values of constant
coefficient hypoelliptic partial differential operators. _Ultradistributional_
is understood in the framework of Denjoy-Carleman classes, i.e., classes
defined via weight sequences. They require, apart from the normalization
condition $1=M_{0}=M_{1}$ as basic assumptions for $M$ log-convexity,
$(\operatorname{mg})$, $(\operatorname{nq})$ and $(\beta_{3})$; see [4, Def.
2.6] (there $(\beta_{3})$ is denoted by $(M.2)^{*}$).

It seems to be reasonable that the results from [4] can be transferred to
$\mathcal{E}_{[\mathcal{M}^{\sigma}]}$: Note that each sequence satisfies all
standard assumptions except $(\operatorname{mg})$ because by $(viii)$ in
Theorem 6.4 and [7, Thm. 3.11] we get $(\beta_{3})$ and even condition
$(M.4)_{a}$ for any $a>0$; see [4, Def. 2.7] resp. [7, Thm. 3.11 $(ii)$].
(Also $M^{(\tau,\tau,\sigma)}_{1}=1$ is violated for $\tau\neq 1$ but which
can be achieved by switching to an equivalent weight matrix.)

Then one can try to compensate the failure of $(\operatorname{mg})$ by
applying $(\mathcal{M}_{\\{\operatorname{mg}\\}})$ resp.
$(\mathcal{M}_{(\operatorname{mg})})$ instead and which is valid by $(iii)$ in
Theorem 6.4.

However, by $(vii)$ in Theorem 5.2 both generalized moderate-growth-type
conditions fail for $\mathcal{M}^{\tau,\sigma}$ and so a generalization of the
proofs from [4] to the PTT-classes
$\mathcal{E}_{[\tau,\sigma]}=\mathcal{E}_{[\mathcal{M}^{\tau,\sigma}]}$ is not
clear. (The other standard properties, except $M^{(c,\tau,\sigma)}_{1}=1$ for
each $c>0$, hold true for each sequence in $\mathcal{M}^{\tau,\sigma}$.)

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