# Approximation classes for adaptive time-stepping finite element methods

Marcelo Actis†, Pedro Morin†, and Cornelia Schneider∗

###### Abstract.

We study approximation classes for adaptive time-stepping finite element
methods for time-dependent Partial Differential Equations (PDE). We measure
the approximation error in $L_{2}([0,T)\times\Omega)$ and consider the
approximation with discontinuous finite elements in time and continuous finite
elements in space, of any degree. As a byproduct we define Besov spaces for
vector-valued functions on an interval and derive some embeddings, as well as
Jackson- and Whitney-type estimates.

###### 2010 Mathematics Subject Classification:

Primary 41A25, 65D05; Secondary 65N30, 65N50

†Partially supported by Agencia Nacional de Promoción Cient\́mathrm{i}fica y
Tecnológica, through grants PICT-2014-2522, PICT-2016-1983, by CONICET through
PIP 2015 11220150100661, and by Universidad Nacional del Litoral through
grants CAI+D 2016-50420150100022LI. A research stay of Pedro Morin at
Universität Erlangen was partially supported by the Simons Foundation and by
the Mathematisches Forschungsinstitut Oberwolfach

∗(Corresponding author) Supported by Deutsche Forschungsgemeinschaft (DFG),
grant SCHN 1509/1-2.

###### Contents

  1. 1 Introduction and main result
  2. 2 Besov spaces of vector-valued functions
    1. 2.1 Moduli of smoothness of vector-valued functions on an interval
    2. 2.2 Besov spaces and embeddings
      1. 2.2.1 Embedding results
  3. 3 Jackson- and Whitney-type theorems for vector-valued functions
    1. 3.1 Jackson’s estimate
    2. 3.2 Whitney’s estimate
  4. 4 Adaptive approximation in one variable
    1. 4.1 The stationary case
    2. 4.2 Greedy algorithm
    3. 4.3 Semi-discretization in time
  5. 5 Discretization in time and space
    1. 5.1 Time marching fully discrete adaptivity
    2. 5.2 Comparison with space-time finite elements

##  1\. Introduction and main result

Adaptive time-stepping finite element methods (AFEM) for evolutionary PDE
usually lead to a sequence of timesteps and meshes, which yield a partition of
the time interval $0=t_{0}<t_{1}<\dots<t_{N}=T$ and one triangulation
$\mathcal{T}_{i}$ for each time interval $[t_{i-1},t_{i})$. The complexity of
the discrete solution is thus related to the total number of degrees of
freedom needed to represent it on the whole interval, which in turn is
equivalent to $\sum_{i=1}^{N}\\#\mathcal{T}_{i}$.

In this article we study spaces of functions which can be approximated using
such time-space partitions with an error of order
$\left(\sum_{i=1}^{N}\\#\mathcal{T}_{i}\right)^{-s}$ for different $s>0$. The
results that we obtain are similar in spirit to those of [BDDP02, GM14], where
the spaces corresponding to stationary PDE are considered.

Our goal is not to prove the optimality of AFEM but rather to understand which
convergence rates are to be expected for the solutions of evolutionary PDE
given their regularity. In this paper we aim at establishing the first results
in this direction, thus at some points we sacrifice generality in order to
have a clearer presentation of the basic ideas and set the foundation for
further research in this area.

In order to roughly state our main result, we need to introduce some notation,
which will be explained in detail later.

Given a polyhedral space domain $\Omega\subset{}^{n}$, $n\geq 1$, we let
$\mathbb{T}$ denote the set of all triangulations that are obtained through
bisection from an initial triangulation $\mathcal{T}_{0}$ of $\Omega$. For
each $\mathcal{T}\in\mathbb{T}$ we denote by $\\#\mathcal{T}$ the number of
elements of the partition

For $\mathcal{T}\in\mathbb{T}$, we let $\mathbb{V}_{\mathcal{T}}^{r}$ denote
the finite element space of continuous piecewise polynomial functions of fixed
order $r$, i.e.,

$\mathbb{V}_{\mathcal{T}}^{r}:=\\{g\in C(\overline{\Omega}):\
g\big{|}_{T}\in\Pi^{r}\ \text{for all }T\in\mathcal{T}\\},$

where $\Pi^{r}$ denotes the set of polynomials of total degree (strictly) less
than $r$.

Let $r_{1},r_{2}\in\mathbb{N}$ denote the polynomial orders in time and space,
respectively. Let $\\{0=t_{0}<t_{1}<\ldots<t_{N}=T\\}$ be a partition of the
time interval and $\mathcal{T}_{1},\ldots,\mathcal{T}_{N}\in\mathbb{T}$ be
partitions of the space domain $\Omega$, where $\mathcal{T}_{i}$ corresponds
to the subinterval $[t_{i-1},t_{i})$, $i=1,\ldots N$. The time-space partition
as illustrated in Figure 1 is then given by

$\mathcal{P}=\left(\\{0=t_{0}<t_{1}<\ldots<t_{N}=T\\},\\{\mathcal{T}_{1},\ldots,\mathcal{T}_{N}\\}\right)\quad\text{with}\quad\\#\mathcal{P}=\sum_{i=1}^{N}\\#\mathcal{T}_{i}$

and $\mathbb{P}$ is the set of all those time-space partitions. This is the
precise kind of time-space partitions produced by time-stepping adaptive
methods.

Figure 1.  Time-space partition $\mathcal{P}$

The finite element space $\overline{\mathbb{V}}_{\mathcal{P}}^{r_{1},r_{2}}$
subject to such a partition $\mathcal{P}$ is defined as

$\displaystyle\overline{\mathbb{V}}_{\mathcal{P}}^{r_{1},r_{2}}$
$\displaystyle:=\\{g:[0,T)\times\Omega\rightarrow\real:g_{[t_{i-1},t_{i})\times\Omega}\in\Pi^{r_{1}}\otimes\mathbb{V}_{\mathcal{T}_{i}}^{r_{2}},\text{
for all }i=1,2,\dots,N\\},$

i.e., $g\in\overline{\mathbb{V}}_{\mathcal{P}}^{r_{1},r_{2}}$ if and only if
$g(t,\cdot)\in\mathbb{V}_{\mathcal{T}_{i}}^{r_{2}}$ for all
$t\in[t_{i-1},t_{i})$ and $g(\cdot,x)\big{|}_{[t_{i-1},t_{i})}\in\Pi^{r_{1}}$
for all $x\in\Omega$, and all $i=1,2,\dots,N$. Discrete solutions of adaptive
time-stepping methods, e.g. those which use Discontinuous Galerkin (DG) in
time, belong to spaces of this type.

We define the best $m$-term approximation error by

$\overline{\sigma}_{m}(f)=\inf_{\\#\mathcal{P}\leq
m}\inf_{g\in\overline{\mathbb{V}}_{\mathcal{P}}^{r_{1},r_{2}}}\|f-g\|_{L_{2}([0,T)\times\Omega)}.$

In this article we measure the error in $L_{2}([0,T)\times\Omega)$ and leave
the general case of $L_{p}([0,T),L_{q}(\Omega))$ and other generalizations as
future work.

For $s>0$ we define the approximation class $\overline{\mathbb{A}}_{s}$ as the
set those functions whose best $m$-term approximation error is of order
$m^{-s}$, i.e.,

$\overline{\mathbb{A}}_{s}:=\\{f\in L_{2}([0,T)\times\Omega):\ \exists
c>0\text{ such that }\overline{\sigma}_{m}(f)\leq c\,m^{-s},\ \forall
m\in\mathbb{N}\\}.$

Equivalently, we can define $\overline{\mathbb{A}}_{s}$ through a semi-norm as
follows:

$\overline{\mathbb{A}}_{s}:=\\{f\in L_{2}([0,T)\times\Omega):\
|f|_{\overline{\mathbb{A}}_{s}}<\infty\\}\quad\text{with}\quad|f|_{\overline{\mathbb{A}}_{s}}:=\sup_{m\in\mathbb{N}}m^{s}\,\overline{\sigma}_{m}(f).$

Alternatively, this definition is equivalent to saying that
$f\in\overline{\mathbb{A}}_{s}$ if there is a constant $c$ such that for all
$\varepsilon>0$, there exists a time-space partition $\mathcal{P}$ that
satisfies

$\inf_{g\in\overline{\mathbb{V}}_{\mathcal{P}}^{r_{1},r_{2}}}\|f-g\|_{L_{2}([0,T)\times\Omega)}\leq
c\varepsilon\quad\text{and}\quad\\#\mathcal{P}\leq\varepsilon^{-1/s},$ (1)

and $|f|_{\overline{\mathbb{A}}_{s}}$ is equivalent to the infimum of all
constants $c$ that satisfy (1).

Our main result is stated in terms of Besov spaces, which will be defined in
the next section, and reads as follows.

######  Main Result 1.

Let $0<s_{i}<r_{i}$, $i=1,2$, $0<q_{1}\leq\infty$, $1\leq q_{2}\leq\infty$
with $s_{1}>\big{(}\frac{1}{q_{1}}-\frac{1}{2}\big{)}_{+}$ and
$s_{2}>n(\frac{1}{q_{2}}-\frac{1}{2}\big{)}_{+}$. Then

$B^{s_{1}}_{q_{1},q_{1}}([0,T),L_{2}(\Omega))\cap
L_{2}([0,T),B_{q_{2},q_{2}}^{s_{2}}(\Omega))\subset\overline{\mathbb{A}}_{s}\quad\text{for}\quad
s=\frac{1}{\frac{1}{s_{1}}+\frac{n}{s_{2}}}.$

This result is a consequence of Theorem 25, where, given $f\in
B^{s_{1}}_{q_{1},q_{1}}([0,T),L_{2}(\Omega)\cap
L_{2}([0,T),B_{q_{2},q_{2}}^{s_{2}}(\Omega))$, and $\varepsilon>0$ we
construct a time-space partition $\mathcal{P}$ that satisfies

$\\#\mathcal{P}\leq
c_{1}\varepsilon^{-\big{(}\frac{1}{s_{1}}+\frac{n}{s_{2}}\big{)}}$

and a function $F\in\overline{\mathbb{V}}_{\mathcal{P}}^{r_{1},r_{2}}$ such
that

$\|f-F\|_{L_{2}([0,T)\times\Omega)}\leq
c_{2}\,\varepsilon\,\left[|f|_{B^{s_{1}}_{q_{1},q_{1}}([0,T),L_{2}(\Omega))}+\|f\|_{L_{2}([0,T),B_{q_{2},q_{2}}^{s_{2}}(\Omega))}\right].$

Here $B^{s}_{p,q}(I,X)$ denote Besov spaces of $X$-valued functions with
respective seminorms $|\cdot|_{B^{s}_{p,q}(I,X)}$, cf. Section 2.2. It is
worth noting that in order to determine the largest spaces, integrability
powers $0<p<1$ must be considered. This makes some proofs more complicated
than if we were to consider only $p\geq 1$.

Our construction is performed in two steps. The first one uses a Greedy
algorithm to obtain the partition of the time domain, resorting in a Whitney-
type estimate for vector-valued functions. That is, we interpret functions in
$L_{2}([0,T)\times\Omega)$ as functions from $[0,T)$ into $L_{2}(\Omega)$ as
is customary in the study of evolutionary PDE, and develop a nonlinear
approximation theory for this situation, by revisiting and extending some
results from Storozhenko and Oswald [Sto77, OS78]. This is presented in
Section 3, after defining Besov spaces of vector-valued functions in Section
2. In Section 4 we revisit the known results for the stationary case and
perform the aforementioned first step by applying the Greedy algorithm to
vector-valued functions. In Section 5 we combine those two results and prove
our main result. We end this article presenting some discussion and comparison
of the approximation classes for space-time discretizations.

We finally mention that we will use $A\lesssim B$ inside some statements,
proofs and reasonings in order to denote $A\leq cB$ with a constant $c$ that
depends on the parameters indicated in the corresponding statement. As usual,
$A\simeq B$ means $A\lesssim B$ and $B\lesssim A$.

##  2\. Besov spaces of vector-valued functions

The goal of this section is to define and understand some properties of Besov
spaces of functions from a real interval $I$ into a Banach space. From now on,
we let $X$ be a separable Banach space with norm $\|\cdot\|_{X}$.

We first introduce the moduli of smoothness and state and prove some of their
properties, which are analogous to those corresponding to the case of real-
valued functions. Afterwards we define the corresponding Besov spaces and
state and prove some embeddings.

###  2.1. Moduli of smoothness of vector-valued functions on an interval

We start this section by providing new definitions of moduli of smoothness for
vector-valued functions, which are analogous to the ones already known for
real-valued functions, and stating and proving some of their basic properties.

It is worth mentioning that there is a forerunner regarding moduli of
smoothness and Whitney-type estimates of vector-valued functions, cf. [DF90].
However, our definition (which is an immediate generalization of the classical
moduli of real-valued functions) differs from the one given in [DF90] (which
is more elaborate and tricky). In particular, in [DF90] a duality approach is
used between the given Banach space and its dual in order to reduce the
definitions and results for abstract functions to real ones. But there is a
price to pay: the results are restricted to the set of bounded functions.
Therefore even classical Banach spaces like $L_{p}(I,X)$ cannot be considered
entirely.

Given $0<p\leq\infty$, a real interval $I$$=[a,b)$ with $|I|=b-a$, and a
function $f:I\to X$, we say that $f\in L_{p}(I,X)$ if $f$ is measurable and
$\|f\|_{L_{p}(I,X)}:=\Big{(}\int_{I}\|f(t)\|_{X}^{p}{\text{d}}t\Big{)}^{1/p}<\infty$
if $p<\infty$ and $\|f\|_{L_{\infty}(I,X)}=\operatorname*{esssup}_{t\in
I}\|f(t)\|_{X}$. For such a function $f$, $r\in\mathbb{N}$ and
$0<|h|<\frac{{|I|}}{r}$, the $r$-th order difference
$\Delta_{h}^{r}f:I_{rh}\to X$ is defined as

$\Delta_{h}^{r}f(t)=\sum_{i=0}^{r}{r\choose i}(-1)^{r-i}f(t+ih),\qquad t\in
I_{rh}:=\\{t\in I:t+rh\in I\\},$

which clearly satisfies $\Delta_{h}^{r}f=\Delta_{h}\Delta_{h}^{r-1}f$ and
$\|\Delta_{h}^{r}f\|_{L_{p}(I_{rh})}^{\min\\{1,p\\}}\leq
2\|\Delta_{h}^{r-1}f\|_{L_{p}(I_{(r-1)h})}^{\min\\{1,p\\}}$, understanding
that $\Delta_{h}f=\Delta_{h}^{1}f$ and $\Delta_{h}^{0}f=f$.

The modulus of smoothness is defined as

$\omega_{r}(f,I,u)_{p}:=\sup_{0<|h|\leq
u}\|\Delta_{h}^{r}f\|_{L_{p}(I_{rh},X)}=\sup_{0<h\leq
u}\|\Delta_{h}^{r}f\|_{L_{p}(I_{rh},X)},\qquad u>0,$ (2)

which is clearly increasing as a function of $u$, and the _averaged_ modulus
of smoothness is defined, for $u>0$, as

$w_{r}(f,I,u)_{p}:=\left(\frac{1}{2u}\int_{-u}^{u}\|\Delta_{h}^{r}f\|^{p}_{L_{p}(I_{rh},X)}\,{\text{d}}h\right)^{\frac{1}{p}}=\left(\frac{1}{u}\int_{0}^{u}\|\Delta_{h}^{r}f\|^{p}_{L_{p}(I_{rh},X)}\,{\text{d}}h\right)^{\frac{1}{p}}.$
(3)

The well-known definitions for $f:\Omega\to\mathbb{R}$, with $\Omega$ a domain
of $\mathbb{R}^{n}$, $n\geq 1$, are as follows. For $h\in\mathbb{R}^{n}$, the
domain of $\Delta_{h}^{r}f$ is the set
$\Omega_{rh}:=\\{x\in\Omega:x,x+h,\dots,x+rh\in\Omega\\}$, and the moduli of
smoothness $\omega_{r}(f,\Omega,u)_{p}$, $w_{r}(t,\Omega,u)_{p}$ are defined
for $u>0$ via

$\displaystyle\omega_{r}(f,\Omega,u)_{p}$ $\displaystyle:=\sup_{0<|h|\leq
u}\|\Delta_{h}^{r}f\|_{L_{p}(\Omega_{rh})},$ (4) $\displaystyle
w_{r}(f,\Omega,u)_{p}$
$\displaystyle:=\left(\frac{1}{(2u)^{n}}\int_{[-u,u]^{n}}\|\Delta_{h}^{r}f\|^{p}_{L_{p}(\Omega_{rh})}\,{\text{d}}h\right)^{\frac{1}{p}}.$

As a consequence of the fact that
$\Delta_{mh}^{1}f(x)=\sum_{i=0}^{m-1}\Delta_{h}^{1}f(x+ih)$, for
$m\in\mathbb{N}$, we can prove by induction
$\|\Delta_{mh}^{r}f\|_{L_{p}(A_{rmh})}\leq
m^{r}\|\Delta_{h}^{r}f\|_{L_{p}(A_{rh})}$, for $A=I$ or $A=\Omega$ (for
details see [PP87, Sect. 3.1]). As an immediate consequence of this,

$\omega_{r}(f,A,mu)_{p}^{\min\\{1,p\\}}\leq
m^{r}\omega_{r}(f,A,u)_{p}^{\min\\{1,p\\}},\quad u>0.$ (5)

From the properties stated above, we have

$w_{r+1}(f,A,u)^{\min\\{1,p\\}}_{p}\leq 2w_{r}(f,A,u)^{\min\\{1,p\\}}_{p}.$
(6)

Finally, we notice that if $f:[a,b)\to X$ and $\hat{f}:[0,1)\to X$ with
$\hat{f}(t)=f(a+t(b-a))$, then, for $u>0$

$\begin{split}\omega_{r}(f,[a,b),u)_{p}&=(b-a)^{1/p}\,\omega_{r}(\hat{f},[0,1),(b-a)^{-1}u)_{p},\\\
w_{r}(f,[a,b),u)_{p}&=(b-a)^{1/p}\,w_{r}(\hat{f},[0,1),(b-a)^{-1}u)_{p}.\end{split}$
(7)

Now we prove that the two moduli of smoothness $w_{r}$ and $\omega_{r}$ as
defined above in (2) and (3) are equivalent. This result is well-known and
proved for real-valued functions in [DL93, Lem. 6.5.1]. The proof for vector-
valued functions is analogous and we sketch it here for completeness.

######  Lemma 1.

Given $0<p<\infty$ and $r\in\mathbb{N}$ the two definitions of moduli of
smoothness $w_{r}(\cdot,\cdot,\cdot)_{p}$ and
$\omega_{r}(\cdot,\cdot,\cdot)_{p}$ are equivalent, more precisely

$\displaystyle w_{r}(f,I,u)_{p}\leq\omega_{r}(f,I,u)_{p}\leq
cw_{r}(f,I,u)_{p},$

for all $f\in L_{p}(I,X)$, $I=[a,b)$ and $0<u<|I|/r$, where the constant $c$
depends only on $r$ and $p$, but is otherwise independent of $f$, $I$, and
$u$.

###### Proof.

The fact that $w_{r}(f,I,u)_{p}\leq\omega_{r}(f,I,u)_{p}$ is obvious.
Therefore, it remains to prove the converse inequality. We prove the result
for the reference situation of $I=[0,1)$, the general case follows by scaling
using (7).

We use the reproducing formula

$\Delta_{h}^{r}f(t)=\sum_{l=1}^{r}(-1)^{l}{r\choose
l}\left[\Delta_{ls}^{r}f(t+lh)-\Delta_{h+ls}^{r}f(t)\right],$ (8)

which holds if $t\in[0,1-rh]$ and

$t+lh+rls\leq 1\quad\text{and}\quad t+rh+rls\leq 1.$

This together yields the range $t\in[0,1-rh-r^{2}s]$. Formula (8) is proved by
induction, starting with the observation that

$\displaystyle\Delta^{1}_{h}f(t)$ $\displaystyle=f(t+h)-f(t)$
$\displaystyle=f(t+h)-f(t+h+ls)+f(t+h+ls)-f(t)$
$\displaystyle=-\big{[}\Delta^{1}_{ls}f(t+h)-\Delta^{1}_{h+ls}f(t)\big{]}.$

We now consider $0<h\leq u\leq\frac{1}{4r}$ and $0\leq t\leq\frac{1}{2}$. This
gives us the upper bound $s<\frac{1}{4r^{2}}$. Integrating formula (8) yields

$\int_{0}^{1/2}\|\Delta^{r}_{h}f(t)\|_{X}^{p}\mathrm{d}t\lesssim\sum_{l=1}^{r}\int_{0}^{1/2}\|\Delta_{ls}^{r}f(t+lh)\|_{X}^{p}\mathrm{d}t+\int_{0}^{1/2}\|\Delta^{r}_{h+ls}f(t)\|_{X}^{p}\mathrm{d}t.$

Thus, setting $I_{-}:=[0,1/2]$ and averaging over $s\in\left[0,u\right]$ gives

$\displaystyle\|\Delta^{r}_{h}f$ $\displaystyle\|_{L_{p}(I_{-},X)}^{p}$ (9)
$\displaystyle\lesssim\sum_{l=1}^{r}\frac{1}{u}\left[\int_{0}^{u}\int_{I_{-}}\|\Delta_{ls}^{r}f(t+lh)\|_{X}^{p}\mathrm{d}t\mathrm{d}s+\int_{0}^{u}\int_{I_{-}}\|\Delta_{h+ls}^{r}f(t)\|_{X}^{p}\mathrm{d}t\mathrm{d}s\right]$
$\displaystyle=\sum_{l=1}^{r}\frac{1}{lu}\Bigg{[}\int_{0}^{lu}\int_{I_{-}}\|\Delta_{h^{\prime}}^{r}f(t+lh)\|_{X}^{p}\mathrm{d}t\mathrm{d}h^{\prime}+\int_{h}^{h+lu}\int_{I_{-}}\|\Delta_{h^{\prime}}^{r}f(t)\|_{X}^{p}\mathrm{d}t\mathrm{d}h^{\prime}\Bigg{]}$
$\displaystyle\lesssim\sum_{l=1}^{r}\frac{1}{(r+1)u}\int_{0}^{(r+1)u}\|\Delta_{h^{\prime}}^{r}f\|_{L_{p}(I,X)}^{p}\mathrm{d}h^{\prime}$
$\displaystyle\leq w_{r}(f,I,(r+1)u)_{p}^{p},$ (10)

where in the second step we used the substitution $h^{\prime}:=ls$ in the
first and $h^{\prime}:=h+ls$ in the second integral. By symmetry, we also have
that $\|\Delta^{r}_{-h}f\|_{L_{p}(I_{+},X)}^{p}\leq w_{r}(f,I,(r+1)u)_{p}^{p}$
with $I_{+}=[1/2,1]$. Taking the supremum w.r.t. $0<h\leq u$ on both sides we
arrive at

$\displaystyle\omega_{r}(f,I,u)_{p}$ $\displaystyle\lesssim
w_{r}(f,I,(r+1)u)_{p}$

Using (5) we obtain

$\displaystyle\omega_{r}(f,I,(r+1)u)_{p}\lesssim\omega_{r}(f,I,u)_{p}\lesssim
w_{r}(f,I,(r+1)u)_{p},$

which completes the proof. ∎

###  2.2. Besov spaces and embeddings

Using the generalized modulus of smoothness defined in the previous
subsection, we introduce the Besov spaces $B^{s}_{p,q}(I,X)$, $s>0$,
$0<p,q\leq\infty$, which contain all functions $f\in L_{p}(I,X)$ such that for
$r:=\lfloor s\rfloor+1$ the quasi-seminorm

$\begin{split}|f|_{B^{s}_{p,q}(I,X)}&:=\displaystyle\left(\int_{0}^{|I|/r}\left[u^{-s}\omega_{r}(f,I,u)_{p}\right]^{q}\frac{\mathrm{d}u}{u}\right)^{1/q}<\infty,\qquad
0<q<\infty,\\\
|f|_{B^{s}_{p,\infty}(I,X)}&:=\sup_{0<u<|I|/r}u^{-s}\omega_{r}(f,I,u)_{p}<\infty.\end{split}$
(11)

Moreover, a quasi-norm for $B^{s}_{p,q}(I,X)$ is given by

$\|f\|_{B^{s}_{p,q}(I,X)}:=\|f\|_{L_{p}(I,X)}+|f|_{B^{s}_{p,q}(I,X)},$ (12)

which is a norm whenever $1\leq p,q\leq\infty$.

######  Remark 2.

One can replace the integral $\int_{0}^{|I|/r}$ by $\int_{0}^{1}$ if
$|I|<\infty$ and still get an equivalent norm. More precisely,

$\int_{0}^{|I|/r}[u^{-s}\omega_{r}(f,I,u)_{p}]^{q}\frac{\mathrm{d}u}{u}\simeq\int_{0}^{1}[u^{-s}\omega_{r}(f,I,u)_{p}]^{q}\frac{\mathrm{d}u}{u}$

with equivalence constants that depend only on $s$, $r$, $p$, $q$, but are
otherwise independent of $f$ and $|I|$ as $|I|\to 0$.

We prove this claim for $0<q<\infty$, the case $q=\infty$ is analogous. If
$|I|/r<1$ then, on the one hand,
$\int_{0}^{|I|/r}[u^{-s}\omega_{r}(f,I,u)_{p}]^{q}\frac{\mathrm{d}u}{u}\leq\int_{0}^{1}[u^{-s}\omega_{r}(f,I,u)_{p}]^{q}\frac{\mathrm{d}u}{u}$.
On the other hand, $\omega_{r}(f,I,u)_{p}=\omega_{r}(f,I,|I|/r)_{p}$, when
$u\geq|I|/r$. Therefore, using (5) and the monotonicity of
$\omega_{r}(f,I,\cdot)_{p}$,

$\displaystyle\int_{|I|/r}^{1}[u^{-s}\omega_{r}(f,I,u)_{p}]^{q}\frac{\mathrm{d}u}{u}$
$\displaystyle=\omega_{r}(f,I,|I|/r)_{p}^{q}\int_{|I|/r}^{1}u^{-sq-1}\mathrm{d}u$
$\displaystyle\lesssim\omega_{r}(f,I,|I|/(2r))_{p}^{q}\,(|I|/r)^{-sq}$
$\displaystyle\lesssim\omega_{r}(f,I,|I|/(2r))_{p}^{q}\int_{|I|/(2r)}^{|I|/r}u^{-sq-1}\mathrm{d}u$
$\displaystyle\leq\int_{|I|/(2r)}^{|I|/r}[u^{-s}\omega_{r}(f,I,u)_{p}]^{q}\frac{\mathrm{d}u}{u},$

which yields the second inequality for the case $|I|/r<1$.

If $|I|/r>1$, trivially
$\int_{0}^{1}[u^{-s}\omega_{r}(f,I,u)_{p}]^{q}\frac{\mathrm{d}u}{u}\leq\int_{0}^{|I|/r}[u^{-s}\omega_{r}(f,I,u)_{p}]^{q}\frac{\mathrm{d}u}{u}$.
Besides, using again (5) and the monotonicity of $\omega_{r}(f,I,\cdot)_{p}$,

$\omega_{r}\left(f,I,\frac{1}{2}\right)_{p}\leq\omega_{r}(f,I,u)_{p}\leq\omega_{r}\left(f,I,\frac{|I|}{r}\right)_{p}\lesssim{|I|^{r}}\omega_{r}\left(f,I,\frac{1}{2}\right)_{p},\quad\frac{1}{2}\leq
u\leq|I|/r.$

Hence,

$\int_{1}^{|I|/r}[u^{-s}\omega_{r}(f,I,u)_{p}]^{q}\frac{\mathrm{d}u}{u}\lesssim{|I|^{rq}}\omega_{r}\left(f,I,\frac{1}{2}\right)_{p}^{q}\lesssim{|I|^{rq}}\int_{\frac{1}{2}}^{1}[u^{-s}\omega_{r}(f,I,u)_{p}]^{q}\frac{\mathrm{d}u}{u},$

which proves the claim for the case $|I|/r>1$.

######  Remark 3.

Our definition for the Besov spaces above is in good agreement with the
standard case: When $f:\Omega\rightarrow\real$, with $\Omega$ a domain of n,
the usual Besov spaces $B^{s}_{p,q}(\Omega)$ are defined as those subspaces
containing all functions $f\in L_{p}(\Omega)$ for which

$|f|_{B^{s}_{p,q}(\Omega)}:=\displaystyle\left(\int_{0}^{\operatorname{diam}(\Omega)}\left[u^{-s}\omega_{r}(f,\Omega,u)_{p}\right]^{q}\frac{\mathrm{d}u}{u}\right)^{1/q}<\infty$
(13)

(with the usual modification if $q=\infty$) and $r=\lfloor s\rfloor+1$. Here
the modulus of smoothness involved is the usual one given in (4). The space
$B^{s}_{p,q}(\Omega)$ is then quasi-normed via
$\|f\|_{B^{s}_{p,q}(\Omega)}:=\|f\|_{L_{p}(\Omega)}+|f|_{B^{s}_{p,q}(\Omega)}.$
For more information on these spaces we refer to [DL93, Tri83].

Later on it will be useful for us to discretize the quasi-seminorm (11) as
follows.

######  Lemma 4.

The quasi-seminorm (11) for $B^{s}_{p,q}(I,X)$ is equivalent to

$\begin{split}|f|_{B^{s}_{p,q}(I,X)}^{*}&:=\left(\sum_{k=0}^{\infty}\left[2^{ks}\omega_{r}(f,I,2^{-k})_{p}\right]^{q}\right)^{1/q},\qquad
0<q<\infty,\\\ |f|_{B^{s}_{p,\infty}(I,X)}^{*}&:=\sup_{k\geq
0}2^{ks}\omega_{r}(f,I,2^{-k})_{p},\end{split}$ (14)

with constants of equivalence independent of $f$ and $I$ as $|I|\to 0$.

###### Proof.

The proof follows along the lines of the standard case, which may be found in
[DL93, p. 56]. Using (5) with $m=2$ and the monotonicity of
$\omega_{r}(f,I,\cdot)$ we see that for $u\in[2^{-k-1},2^{-k}]$ it holds

$\displaystyle
2^{-r}\left(2^{ks}\omega_{r}(f,I,2^{-k})_{p}\right)^{\min\\{1,p\\}}$
$\displaystyle\leq\left(u^{-s}\omega_{r}(f,I,u)\right)^{\min\\{1,p\\}}_{p}$
$\displaystyle\leq\left(2^{(k+1)s}\omega_{r}(f,I,2^{-k})_{p}\right)^{\min\\{1,p\\}}.$

Raising all terms of the inequality to the power ${\frac{1}{\min\\{1,p\\}}}$
we obtain

$u^{-s}\omega_{r}(f,I,u)_{p}\simeq
2^{ks}\omega_{r}(f,I,2^{-k})_{p}\qquad\text{for}\qquad u\in[2^{-k-1},2^{-k}].$

Hence, since $\int_{2^{-k-1}}^{2^{-k}}\frac{\mathrm{d}u}{u}=\ln 2\simeq 1$ we
get

$\left(\int_{2^{-k-1}}^{2^{-k}}[u^{-s}\omega_{r}(f,I,u)_{p}]^{q}\frac{\mathrm{d}u}{u}\right)^{1/q}\simeq
2^{ks}\omega_{r}(f,I,2^{-k})_{p}.$

This completes the proof for $0<q<\infty$ taking into account Remark 2, after
adding all terms for $k\geq 0$. The case $q=\infty$ is analogous. ∎

####  2.2.1. Embedding results

Before we provide some embeddings for the scale $B^{s}_{p,q}(I,X)$ needed
later on, let us briefly recall what is known concerning the Besov spaces
$B^{s}_{p,q}(\Omega)$.

######  Proposition 5.

Let $s>0$ and $0<p,q\leq\infty$.

  * (i)

Let $0<\varepsilon<s$, $0<\nu\leq\infty$, and $q\leq\vartheta\leq\infty$, then

$B^{s}_{p,q}(\Omega)\hookrightarrow
B^{s-\varepsilon}_{p,\nu}(\Omega)\qquad\text{and}\qquad
B^{s}_{p,q}(\Omega)\hookrightarrow B^{s}_{p,\vartheta}(\Omega).$

  * (ii)

(Sobolev-type embedding) Let $0<\sigma<s$ and $p<\tau$ be such that

$s-\frac{n}{p}\geq\sigma-\frac{n}{\tau},$ (15)

then

$B^{s}_{p,q}(\Omega)\hookrightarrow B^{\sigma}_{\tau,\vartheta}(\Omega),$ (16)

where $0<\vartheta\leq\infty$ and, additionally, $q\leq\vartheta$ if an
equality holds in (15). Moreover, in the limiting case when $\sigma=0$ and
$\vartheta$ is such that

$s-\frac{n}{p}\geq-\frac{n}{\vartheta},$ (17)

we have

$B^{s}_{p,q}(\Omega)\hookrightarrow L_{\vartheta}(\Omega),$ (18)

where again $q\leq\vartheta$ if an equality holds in (17).

  * (iii)

If the domain $\Omega\subset\mathbb{R}^{n}$ is bounded, then for $\tau\leq p$
we have the embedding

$B^{s}_{p,q}(\Omega)\hookrightarrow B^{s}_{\tau,q}(\Omega).$ (19)

######  Remark 6.

  * (i)

The above results can be found in [DL93, § 2.10, 12.8], [HS09, Thm. 1.15], and
[BS88].
In the interpolation diagram aside we have illustrated the area of possible
embeddings of a fixed original space $B^{s}_{p,q}(\Omega)$ into spaces
$B^{\sigma_{1}}_{\tau_{1},\nu_{1}}(\Omega)$ and
$B^{\sigma_{2}}_{\tau_{2},\nu_{2}}(\Omega)$. The lighter shaded area
corresponds to the additional embeddings we have if the underlying domain
$\Omega$ is bounded.

Figure 2. Embeddings for $B^{s}_{p,q}(\Omega)$

  * (ii)

In the non-limiting case (corresponding to the strict inequality in (15) and
(17)) the embeddings in Proposition 5 are known to be compact. In particular,
for $\alpha>0$ and $p<\tau$, the embeddings
$B^{s+\alpha}_{p,p}(\Omega)\hookrightarrow B^{s}_{\tau,\tau}(\Omega)$ ($s>0$)
and $B^{\alpha}_{p,p}(\Omega)\hookrightarrow L_{\tau}(\Omega)$ ($s=0$) are
compact if, and only if,

$\alpha-\frac{n}{p}>-\frac{n}{\tau}.$

For the scale $B^{s}_{p,q}(I,X)$ there are counterparts of the embeddings from
Proposition 5.

######  Proposition 7.

Assume $s>0$ and $0<p,q\leq\infty$.

  * (i)

Let $0<\varepsilon<s$, $0<\nu\leq\infty$, and $q\leq\vartheta\leq\infty$, then

$B^{s}_{p,q}(I,X)\hookrightarrow
B^{s-\varepsilon}_{p,\nu}(I,X)\qquad\text{and}\qquad
B^{s}_{p,q}(I,X)\hookrightarrow B^{s}_{p,\vartheta}(I,X).$

  * (ii)

If the time interval $I$ is bounded, then for $\tau\leq p$ we have the
embedding

$B^{s}_{p,q}(I,X)\hookrightarrow B^{s}_{\tau,q}(I,X).$ (20)

###### Proof.

The embeddings in (i) and (ii) can be proven as in the standard case, using
the discrete version of the seminorm for Besov spaces, i.e.,

$|f|_{B^{s}_{p,q}(I,X)}\simeq|f|_{B^{s}_{p,q}(I,X)}^{*}=\left(\sum_{k=0}^{\infty}[2^{ks}\omega_{r}(f,2^{-k})_{p}]^{q}\right)^{\frac{1}{q}},\quad
0<q<\infty,$

with the analogous one for $q=\infty$. Indeed, the second embedding in (i) is
just a consequence of the monotonicity of the $\ell_{q}$ sequence spaces,
i.e., $\ell_{q}\hookrightarrow\ell_{\vartheta}$ for $q\leq\vartheta$. The
first embedding for $q\leq\nu$ is also clear since $2^{k(s-\varepsilon)}\leq
2^{ks}$. If $\nu<q$ one uses Hölder’s inequality with $\frac{q}{\nu}>1$, which
gives the desired result.
Moreover, (ii) follows immediately since for $\tau\leq p$ and $|I|<\infty$ we
have $L_{p}(I,X)\hookrightarrow L_{\tau}(I,X).$ ∎

######  Remark 8.

The counterpart of the limiting embedding (18) in Prop. 5(ii) is derived in
Corollary 21 as an application of our generalized Whitney’s estimate presented
in Proposition 20. Moreover, the Sobolev-type embeddings as stated in Prop.
5(ii), formula (16), should also hold. The proof in the standard case, cf.
[DL93, § 12.8], involves spline representations for Besov spaces, which we
have not provided for our generalized setting so far. This is out of the scope
of the present paper.

##  3\. Jackson- and Whitney-type theorems for vector-valued functions

In this section we prove Jackson- and Whitney-type theorems for functions
defined on an interval, but valued on a Banach space. Some proofs are rather
technical, and analogous to the ones presented for scalar-valued functions in
[Sto77, OS78].

Let us mention that regarding Jackson’s theorem there is a proof for $1\leq
p\leq\infty$, which is based on the $K$-functional method of interpolation
[PP87, §3.5] and seems extendable to vector-valued functions. There is an
alternative proof in [PP87, Thm. 7.1], which holds for $0<p\leq\infty$ and
avoids all the technicalities from [Sto77, OS78]. However, it is based on a
contradiction argument and does not work in the vector-valued case, or at
least we could not generalize it to the infinite-dimensional setting.

The proof of Whitney’s theorem that we present below in Section 3.2 follows
the steps from [DeV98, Sect. 6.1]. In order to do it, we need an equivalence
of $L_{p}$-norms for vector-valued polynomials, which is contained in Lemma 17
and Corollary 19. After proving Whitney’s estimate in $B^{s}_{q,q}(I,X)\cap
L_{p}(I,X)$ in Proposition 20 we obtain the embedding $B^{s}_{q,q}(I,X)\subset
L_{p}(I,X)$, and arrive at Whitney’s estimate in $B^{s}_{q,q}(I,X)$.

###  3.1. Jackson’s estimate

The goal of this section is to prove a Jackson-type estimate, which is stated
below in Theorem 9 and requires some definitions.

Given a separable Banach space $X$, $r\in\mathbb{N}$, and an interval
$I=[a,b)$, we denote by $\mathbb{V}^{r}_{I,X}$ the space of $X$-valued
polynomials of order $r$ w.r.t. time, which we define as follows:

$\mathbb{V}^{r}_{I,X}:=\bigg{\\{}P:I\to
X,P(t)=\sum_{j=1}^{r}\ell_{j}^{r}(t)P_{j}:\ P_{j}\in X,\ t\in I\bigg{\\}},$
(21)

with $\ell_{j}^{r}$ the usual (scalar-valued) Lagrange basis functions

$\ell_{j}^{r}(t)=\prod_{i\neq j}\frac{t-t_{i}}{t_{j}-t_{i}}\qquad\text{for
}t_{j}=a+(j-1)\frac{b-a}{r-1},\qquad j=1,2,\dots,r.$ (22)

Notice that any basis for the space $\Pi^{r}$ of scalar-valued polynomials in
, such as $1,t,t^{2},\dots,t^{r-1}$, leads to the same space
$\mathbb{V}^{r}_{I,X}$.

The main result of this section is the following.

######  Theorem 9 (Jackson’s Theorem).

Let $0<p\leq\infty$ and $r\in\mathbb{N}$. Then there exists a constant $c>0$
such that for any interval $I$ and every $f\in L_{p}(I,X)$, there exists a
vector-valued polynomial $P_{r}\in\mathbb{V}_{I,X}^{r}$, which satisfies

$\|f-P_{r}\|^{p}_{L_{p}(I,X)}\leq
c\,w_{r}(f,I,h)_{p}^{p}\qquad\text{with}\qquad h=\frac{|I|}{2r}.$ (23)

In other words, there exist $a_{0}$, $a_{1}$, …, $a_{r-1}\in X$ such that, if
$P_{r}(t)=a_{0}+a_{1}t+\ldots+a_{r-1}t^{r-1}$, then (23) holds.

Due to the homogeneity (5) and the equivalence of Lemma 1, Jackson’s estimate
can also be stated as:

$E_{r}(f,I)_{p}:=\inf_{P_{r}\in\mathbb{V}_{I,X}^{r}}\|f-P_{r}\|^{p}_{L_{p}(I,X)}\leq
c\,w_{r}(f,I,|I|)_{p}^{p},\qquad\forall f\in L_{p}(I,X).$ (24)

In order to prove this estimate, we need several auxiliary lemmas, which are
rather technical, and analogous to the ones proved for scalar-valued functions
in [Sto77, OS78]. We generalize them to our setting. The basic idea is to
first study periodic functions and their higher order differences, and then
relate them to differences of the functions we are actually interested in.

Let $f:[a,b)\rightarrow X$ be an $X$-valued function and $f^{\ast}$ denote its
periodic continuation with period $d:=b-a$, i.e.,

$f^{\ast}(t)=f(t-\ell d),\quad\text{where $\ell\in\mathbb{Z}$ is such that
$t-\ell d\in[a,b)$}.$

Moreover, for $0<p<\infty$ and $k\in\mathbb{N}$ consider the integrals

$\displaystyle I^{\ast}_{p,k}(h)$
$\displaystyle:=\int_{a}^{b}\|\Delta^{k}_{h}f^{\ast}(t)\|_{X}^{p}\mathrm{d}t=\int_{0}^{d}\|\Delta^{k}_{h}f^{\ast}(t)\|_{X}^{p}\mathrm{d}t,$
(25) $\displaystyle I_{p,k}(h)$
$\displaystyle:=\int_{a}^{b-kh}\|\Delta^{k}_{h}f(t)\|_{X}^{p}\mathrm{d}t.$
(26)

Note that we do not emphasize on the fact that the expressions
$I^{\ast}_{p,k}(h)$ and $I_{p,k}(h)$ also depend on the functions $f$ and
$f^{\ast}$, respectively, since it will always be clear from the context which
function we deal with.

We start with the following result showing how the best approximation of some
function $f\in L_{p}(I,X)$ by a constant $a_{0}\in X$ can be bounded using
first differences of its periodic continuation $f^{\ast}$.

######  Lemma 10.

Let $0<p<\infty$ and $f\in L_{p}(I,X)$. There exists $a_{0}\in X$ such that

$\|f-a_{0}\|^{p}_{L_{p}(I,X)}\leq\frac{1}{d}\int_{0}^{d}I^{\ast}_{p,1}(y)\mathrm{d}y.$

###### Proof.

We show how to construct $a_{0}\in X$ satisfying the desired inequality. Let
$f^{\ast}$ denote the $d$-periodic continuation of $f$. We make the following
easy observation,

$\displaystyle\inf_{a_{0}\in X}\|f-a_{0}\|_{L_{p}(I,X)}^{p}$
$\displaystyle=\inf_{a_{0}\in
X}\int_{0}^{d}\|f^{\ast}(t)-a_{0}\|_{X}^{p}\mathrm{d}t$
$\displaystyle=\inf_{a_{0}\in
X}\int_{0}^{d}\|f^{\ast}(t+y)-a_{0}\|_{X}^{p}\mathrm{d}t$
$\displaystyle\leq\int_{0}^{d}\|f^{\ast}(t+y)-f^{\ast}(y)\|_{X}^{p}\mathrm{d}t,\quad\text{for
any $y\in[0,d)$}.$

Now using the fact that $f^{\ast}$ is $d$-periodic and the left-hand side does
not depend on $y$, integration from $0$ to $d$ w.r.t. $y$ yields

$\displaystyle\inf_{a_{0}\in X}\|f-a_{0}\|_{L_{p}(I,X)}^{p}$
$\displaystyle\leq\frac{1}{d}\int_{0}^{d}\int_{0}^{d}\|f^{\ast}(t+y)-f^{\ast}(y)\|_{X}^{p}\mathrm{d}t\mathrm{d}y$
$\displaystyle=\frac{1}{d}\int_{a}^{b}\int_{a}^{b}\|f(t)-f(y)\|_{X}^{p}\mathrm{d}t\mathrm{d}y=\frac{1}{d}\int_{a}^{b}g(y)\mathrm{d}y,$

where in the last line we put
$g(y):=\int_{a}^{b}\|f(t)-f(y)\|_{X}^{p}\mathrm{d}t$. Note that the set $S$
defined as

$S:=\Big{\\{}z\in[a,b):\
g(z)\leq\frac{1}{d}\int_{a}^{b}g(y)\mathrm{d}y\Big{\\}},$

is non-empty. Therefore, taking $z\in S$ and putting $a_{0}:=f(z)$ we obtain

$\displaystyle\|f-a_{0}\|^{p}_{L_{p}(I,X)}$
$\displaystyle=\int_{a}^{b}\|f(t)-f(z)\|_{X}^{p}\mathrm{d}t=g(z)$
$\displaystyle\leq\frac{1}{d}\int_{a}^{b}\int_{a}^{b}\|f(t)-f(y)\|_{X}^{p}\mathrm{d}t\mathrm{d}y$
$\displaystyle=\frac{1}{d}\int_{0}^{d}\int_{0}^{d}\|f^{\ast}(t+y)-f^{\ast}(y)\|_{X}^{p}\mathrm{d}t\mathrm{d}y$
$\displaystyle=\frac{1}{d}\int_{0}^{d}\int_{0}^{d}\|f^{\ast}(t+y)-f^{\ast}(y)\|_{X}^{p}\mathrm{d}y\mathrm{d}t$
$\displaystyle=\frac{1}{d}\int_{0}^{d}I^{\ast}_{p,1}(t)\mathrm{d}t,$

which shows that $a_{0}:=f(z)$ with $z\in S$ yields the assertion. ∎

The following lemma shows that we can bound integrals of lower order
differences of periodic functions with integrals involving higher order
differences.

######  Lemma 11.

Let $0<p<\infty$ and $k\in\mathbb{N}$. Then we have the following relation

$\int_{0}^{d}I^{\ast}_{p,k}(y)\mathrm{d}y\leq
c\,\int_{0}^{d}I^{\ast}_{p,k+1}(y)\mathrm{d}y,$

with the constant $c>0$ only depending on $k$ and $p$, but otherwise
independent of the function $f$ and the interval $[a,b)$.

###### Proof.

We make use of the following identity

$\Delta^{k}_{2y}f^{\ast}(t)-2^{k}\Delta^{k}_{y}f^{\ast}(t)=\sum_{i=1}^{k}{k\choose
i}\sum_{m=0}^{i-1}\Delta_{y}^{k+1}f^{\ast}(t+my),$ (27)

which can be found in [Tim63, Sect. 3.3.2]. Let $0<p<1$. In this case we know
that $|\cdot|^{p}$ is subadditive. This and integration from $0$ to $d$ w.r.t.
$t$ in (27) leads to

$2^{kp}\int_{0}^{d}\|\Delta_{y}^{k}f^{\ast}(t)\|_{X}^{p}\mathrm{d}t-\int_{0}^{d}\|\Delta^{k}_{2y}f^{\ast}(t)\|_{X}^{p}\mathrm{d}t\\\
\leq\sum_{i=1}^{k}{k\choose
i}\sum_{m=0}^{i-1}\int_{0}^{d}\|\Delta_{y}^{k+1}f^{\ast}(t+my)\|_{X}^{p}\mathrm{d}t.$

Now integrating once more from $0$ to $d$ w.r.t. $y$ and using the definition
of $I^{\ast}_{p,k}$ gives

$2^{kp}\int_{0}^{d}I^{\ast}_{p,k}(y)\mathrm{d}y-\int_{0}^{d}I_{p,k}^{*}(2y)\mathrm{d}y\leq\sum_{i=1}^{k}{k\choose
i}\,i\int_{0}^{d}I_{p,k+1}^{\ast}(y)\mathrm{d}y.$ (28)

Since $I^{\ast}_{p,k}(y)$ is $d$-periodic, we have the identity

$\int_{0}^{d}I^{\ast}_{p,k}(2y)\mathrm{d}y=\frac{1}{2}\int_{0}^{2d}I^{\ast}_{p,k}(y^{\prime})\mathrm{d}y^{\prime}=\int_{0}^{d}I^{\ast}_{p,k}(y)\mathrm{d}y.$

Inserting this in (28) we obtain

$\left(2^{kp}-1\right)\int_{0}^{d}I^{\ast}_{p,k}(y)\mathrm{d}y\leq
c_{k,p}\int_{0}^{d}I_{p,k+1}^{\ast}(y)\mathrm{d}y,$ (29)

which gives the desired estimate in the case $0<p<1$. When $1\leq p<\infty$ we
proceed with (27) as follows: We add $2^{k}\Delta^{k}_{y}f^{\ast}(t)$ on both
sides of (27) and integrate from $0$ to $d$ w.r.t. $t$ and from $0$ to $d$
w.r.t. $y$ afterwards in the $L_{p}$-norm. This gives (28) but with the
integrals to the power $\frac{1}{p}$. We proceed as before and end up with
(29) to the power $\frac{1}{p}$, which proves the asserted estimate. ∎

The following lemma shows how to bound integrals of higher order differences
of the periodic extension of a function by integrals of higher order
differences of the original function plus first order differences.

######  Lemma 12.

Let $f\in L_{p}(I,X)$, where $0<p<\infty$. Then for any $k\in\mathbb{N}$ it
holds

$\int_{0}^{\frac{d}{k}}I^{\ast}_{p,k}(y)\mathrm{d}y\leq
2\int_{0}^{\frac{d}{k}}I_{p,k}(y)\mathrm{d}y+c\,d\,I_{p,1}\left(\frac{d}{k}\right).$

with the constant $c>0$ only depending on $k$ and $p$, but otherwise
independent of the function $f$ and the interval $[a,b)$.

###### Proof.

By definition $\Delta^{k}_{y}f^{\ast}(t)=\sum_{i=0}^{k}(-1)^{k-i}{k\choose
i}f^{\ast}(t+iy)$ and the fact that $f^{\ast}$ is the $d$-periodic
continuation of $f$, i.e., $f=f^{\ast}$ on $[a,b)$ and $f^{\ast}(t)=f(t-d)$
for some $t\in[b,b+d)$, we express $I_{p,k}^{\ast}$ in terms of the values of
$f$ as follows:

$\displaystyle I^{\ast}_{p,k}(y)$
$\displaystyle=\int_{a}^{b}\|\Delta_{y}^{k}f^{\ast}(t)\|_{X}^{p}\mathrm{d}t$
$\displaystyle=\int_{a}^{b-ky}\|\Delta_{y}^{k}f(t)\|_{X}^{p}\mathrm{d}t+\sum_{j=1}^{k}\int_{b-jy}^{b-(j-1)y}\|S_{j}\|_{X}^{p}\mathrm{d}t,\qquad
0\leq y\leq\frac{d}{k},$ (30)

where

$S_{j}(t)=\sum_{i=0}^{j-1}(-1)^{k-i}{k\choose
i}f(t+iy)+\sum_{i=j}^{k}(-1)^{k-i}{k\choose i}f(t+iy-d).$

Figure 3. Express $f^{\ast}$ via $f$ with $j=3$

Now we transform $S_{j}$ as follows: we augment the first term of the first
sum and the last of the second sum in order to obtain the value of the $k$-th
difference of $f$ at the point $t$ with step $y-\frac{d}{k}$. This yields

$\displaystyle S_{j}$ $\displaystyle=\sum_{i=0}^{k}(-1)^{k-i}{k\choose
i}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}f\left(t+i\Big{(}y-\frac{d}{k}\Big{)}\right)}\quad{{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\nearrow
i=0\text{ first sum
\quad}}\atop{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\searrow
i=k\text{ second sum}}}$ $\displaystyle(=:T_{1})$
$\displaystyle\quad+{\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}\sum_{i=1}^{j-1}}(-1)^{k-i}{k\choose
i}\left[f\left(t+iy\right){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-f\left(t+i\Big{(}y-\frac{d}{k}\Big{)}\right)}\right]$
$\displaystyle(=:T_{2}(j))$
$\displaystyle\quad+{\color[rgb]{0,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,1}\pgfsys@color@cmyk@stroke{1}{0}{0}{0}\pgfsys@color@cmyk@fill{1}{0}{0}{0}\sum_{i=j}^{k-1}}(-1)^{k-i}{k\choose
i}\left[f\left(t+iy-d\right){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-f\left(t+i\Big{(}y-\frac{d}{k}\Big{)}\right)}\right]$
$\displaystyle(=:T_{3}(j))$ $\displaystyle=:T_{1}+T_{2}(j)+T_{3}(j).$

Since $T_{1}$ does not depend on $j$,

$\displaystyle\sum_{j=1}^{k}\int_{b-jy}^{b-(j-1)y}\|T_{1}\|_{X}^{p}\,\mathrm{d}t$
$\displaystyle=\int_{b-ky}^{b}\|T_{1}\|_{X}^{p}\mathrm{d}t=\int_{b-ky}^{b}\|\Delta^{k}_{y-\frac{d}{k}}f(t)\|_{X}^{p}\,\mathrm{d}t$
$\displaystyle=\int_{a}^{a+ky}\|\Delta^{k}_{\frac{d}{k}-y}f(t)\|_{X}^{p}\mathrm{d}t=I_{p,k}\left(\frac{d}{k}-y\right),$
(31)

where in the third step we changed the step $y-\frac{d}{k}$ involving the
$k$-th difference of $f$ into $\frac{d}{k}-y$ in order to obtain a nonnegative
step. We now estimate the sum

$\displaystyle\sum_{j=1}^{k}$
$\displaystyle\int_{b-jy}^{b-(j-1)y}\|T_{2}(j)\|_{X}^{p}+\|T_{3}(j)\|_{X}^{p}\mathrm{d}t$
$\displaystyle\lesssim\sum_{j=1}^{k}\int_{b-jy}^{b-(j-1)y}\left\\{\sum_{i=1}^{j-1}{k\choose
i}^{p}\Big{\|}f(t+iy)-f\left(t+i\Big{(}y-\frac{d}{k}\Big{)}\right)\Big{\|}_{X}^{p}\right.$
$\displaystyle\qquad\qquad\qquad\qquad+\left.\sum_{i=j}^{k-1}{k\choose
i}^{p}\Big{\|}f(t+iy-d)-f\left(t+i\Big{(}y-\frac{d}{k}\Big{)}\right)\Big{\|}_{X}^{p}\right\\}\mathrm{d}t$
(change summation
$\sum_{j=1}^{k}\sum_{i=1}^{j-1}=\sum_{i=1}^{k-1}\sum_{j=i+1}^{k}$ and
$\sum_{j=1}^{k}\sum_{i=j}^{k-1}=\sum_{i=1}^{k-1}\sum_{j=1}^{i}$)
$\displaystyle=\sum_{i=1}^{k-1}{k\choose
i}^{p}\int_{b-ky}^{b-iy}\left\|f(t+iy)-f\left(t+i\Big{(}y-\frac{d}{k}\Big{)}\right)\right\|_{X}^{p}\mathrm{d}t$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\sum_{i=1}^{k-1}{k\choose
i}^{p}\int_{b-iy}^{b}\left\|f(t+iy-d)-f\left(t+i\Big{(}y-\frac{d}{k}\Big{)}\right)\right\|_{X}^{p}\mathrm{d}t$
(1st integral: Substitution $t^{\prime}=t+i\left(y-\frac{d}{k}\right)$ ;
reverse sum $i\mapsto k-i$) (2nd integral: Substitution
$t^{\prime\prime}=t+iy-d$) $\displaystyle=\sum_{i=1}^{k-1}{k\choose
i}^{p}\int_{\frac{k-i}{k}a+\frac{i}{k}b-iy}^{\frac{k-i}{k}a+\frac{i}{k}b}\left\|f\left(t^{\prime}+(k-i)\frac{d}{k}\right)-f\left(t^{\prime}\right)\right\|_{X}^{p}\mathrm{d}t^{\prime}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\sum_{i=1}^{k-1}{k\choose
i}^{p}\int_{a}^{a+iy}\left\|f(t^{\prime\prime})-f\left(t^{\prime\prime}+(k-i)\frac{d}{k}\right)\right\|_{X}^{p}\mathrm{d}t^{\prime\prime}.$
(32)

Now (30), (31), and (32) yield

$\displaystyle I^{\ast}_{p,k}(y)\leq{}$ $\displaystyle
I_{p,k}(y)+I_{p,k}\left(\frac{d}{k}-y\right)$
$\displaystyle+\sum_{i=1}^{k-1}{k\choose
i}^{p}\int_{\frac{k-i}{k}a+\frac{i}{k}b-iy}^{\frac{k-i}{k}a+\frac{i}{k}b}\left\|f\left(t^{\prime}+(k-i)\frac{d}{k}\right)-f\left(t^{\prime}\right)\right\|_{X}^{p}\mathrm{d}t^{\prime}$
$\displaystyle+\sum_{i=1}^{k-1}{k\choose
i}^{p}\int_{a}^{a+iy}\left\|f(t^{\prime\prime})-f\left(t^{\prime\prime}+(k-i)\frac{d}{k}\right)\right\|_{X}^{p}\mathrm{d}t^{\prime\prime}.$

Integrating from $0$ to $\frac{d}{k}$ w.r.t. $y$ gives

$\displaystyle\int_{0}^{\frac{d}{k}}I^{\ast}_{p,k}(y)\mathrm{d}y\leq{}$
$\displaystyle\int_{0}^{\frac{d}{k}}I_{p,k}(y)\mathrm{d}y+\int_{0}^{\frac{d}{k}}I_{p,k}\left(\frac{d}{k}-y\right)\mathrm{d}y$
$\displaystyle+c\left\\{\sum_{i=1}^{k-1}\int_{0}^{\frac{d}{k}}\int_{\frac{k-i}{k}a+\frac{i}{k}b-iy}^{\frac{k-i}{k}a+\frac{i}{k}b}\left\|f\left(t^{\prime}+(k-i)\frac{d}{k}\right)-f\left(t^{\prime}\right)\right\|_{X}^{p}\mathrm{d}t^{\prime}\mathrm{d}y\right.$
$\displaystyle\quad+\left.\sum_{i=1}^{k-1}\int_{0}^{\frac{d}{k}}\int_{a}^{a+iy}\left\|f(t^{\prime\prime})-f\left(t^{\prime\prime}+(k-i)\frac{d}{k}\right)\right\|_{X}^{p}\mathrm{d}t^{\prime\prime}\mathrm{d}y\right\\}.$

We change the order of integration in the double integrals. For the second
integral this yields

$\int_{0}^{\frac{d}{k}}\int_{a}^{a+iy}(\ldots)\mathrm{d}t^{\prime\prime}\mathrm{d}y\longrightarrow\int_{a}^{a+i\frac{d}{k}}\int_{\frac{t^{\prime\prime}-a}{i}}^{\frac{d}{k}}(\ldots)\mathrm{d}y\mathrm{d}t^{\prime\prime}.$

Similarly for the first one. Moreover, observing that the integrand in both
cases does not depend on $y$ we obtain

$\displaystyle\int_{0}^{\frac{d}{k}}$ $\displaystyle
I^{\ast}_{p,k}(y)\mathrm{d}y$ $\displaystyle\leq
2\int_{0}^{\frac{d}{k}}I_{p,k}(y)\mathrm{d}y+c\bigg{\\{}\sum_{i=1}^{k-1}\int_{a}^{a+i\frac{d}{k}}\left(\frac{t^{\prime}}{i}-\frac{a}{i}\right)\left\|f\left(t^{\prime}+(k-i)\frac{d}{k}\right)-f\left(t^{\prime}\right)\right\|_{X}^{p}\mathrm{d}t^{\prime}$
$\displaystyle\qquad+\sum_{i=1}^{k-1}\int_{a}^{a+i\frac{d}{k}}\left(\frac{d}{k}-\frac{t^{\prime\prime}-a}{i}\right)\left\|f(t^{\prime\prime})-f\left(t^{\prime\prime}+(k-i)\frac{d}{k}\right)\right\|_{X}^{p}\mathrm{d}t^{\prime\prime}\bigg{\\}}$
$\displaystyle=2\int_{0}^{\frac{d}{k}}I_{p,k}(y)\mathrm{d}y+c\frac{d}{k}\sum_{i=1}^{k-1}\int_{a}^{a+i\frac{d}{k}}\left\|f(t)-f\left(t+(k-i)\frac{d}{k}\right)\right\|_{X}^{p}\mathrm{d}t.$
(33)

Using a telescopic sum we see that

$\displaystyle\Big{\|}f(t)-f\left(t+(k-i)\frac{d}{k}\right)\Big{\|}^{p}_{X}\lesssim\sum_{j=1}^{k-i}\Big{\|}f\left(t+(j-1)\frac{d}{k}\right)-f\left(t+j\frac{d}{k}\right)\Big{\|}^{p}_{X}$

and for $i=1,2,\ldots,k-1$,

$\displaystyle\sum_{j=1}^{k-i}$
$\displaystyle\int_{a}^{a+i\frac{d}{k}}\left\|f\left(t+(j-1)\frac{d}{k}\right)-f\left(t+j\frac{d}{k}\right)\right\|_{X}^{p}\mathrm{d}t$
$\displaystyle=\sum_{j=1}^{k-i}\int_{a+\frac{j-1}{k}d}^{a+\frac{i+j-1}{k}d}\left\|f\left(t^{\prime}\right)-f\left(t^{\prime}+\frac{d}{k}\right)\right\|_{X}^{p}\mathrm{d}t^{\prime}$
$\displaystyle\leq(k-i)\int_{a}^{a+\frac{k-1}{k}d}\left\|f\left(t^{\prime}\right)-f\left(t^{\prime}+\frac{d}{k}\right)\right\|_{X}^{p}\mathrm{d}t^{\prime},$
(34)

where in the second step we used a change of variables
$t^{\prime}:=t+(j-1)\frac{d}{k}$. Inserting (34) into (33) finally gives

$\displaystyle\int_{0}^{\frac{d}{k}}I^{\ast}_{p,k}(y)\mathrm{d}y$
$\displaystyle\leq
2\int_{0}^{\frac{d}{k}}I_{p,k}(y)\mathrm{d}y+d\,c_{k,p}\int_{a}^{b-\frac{d}{k}}\left\|f\left(t^{\prime}\right)-f\left(t^{\prime}+\frac{d}{k}\right)\right\|_{X}^{p}\mathrm{d}t^{\prime}$
$\displaystyle=2\int_{0}^{\frac{d}{k}}I_{p,k}(y)\mathrm{d}y+c_{k,p}\,d\,I_{p,1}\left(\frac{d}{k}\right),$

which completes the proof. ∎

The previous lemmas give the following result, which shows that we can bound
the best approximation of a function $f\in L_{p}(I,X)$ by a constant $a_{0}\in
X$ with the help of integrals of higher order differences and first order
differences of $f$.

######  Lemma 13.

Let $I=[a,b)$, $0<p<\infty$, and $m\in\mathbb{N}$. There exists a constant
$c=c_{m,p}$, such that for every $f\in L_{p}(I,X)$ there exists $a_{0}\in X$
satisfying, for $h=\frac{b-a}{m}$,

$\displaystyle\|f-a_{0}\|^{p}_{L_{p}(I,X)}$ $\displaystyle\leq
c\left[\frac{1}{h}\int_{0}^{h}\int_{a}^{b-ms}\|\Delta_{s}^{m}f(t)\|_{X}^{p}\mathrm{d}t\mathrm{d}s+\int_{a}^{b-h}\|\Delta_{h}f(t)\|_{X}^{p}\mathrm{d}t\right]$
(35)
$\displaystyle=c\left[w_{m}(f,I,h)_{p}^{p}+\|\Delta_{h}f(t)\|^{p}_{L_{p}(I_{h},X)}\right].$

######  Remark 14.

Note that the second term with the first order differences in (35) is crucial:
If $f$ is a polynomial of degree $m-1$ the first integral on the right-hand
side vanishes but the left-hand side might not.

###### Proof.

We first notice that, by induction, we can easily check that

$\Delta_{my}^{m}f^{*}(t)=\sum_{i_{m}=0}^{m-1}\dots\sum_{i_{1}=0}^{m-1}\Delta_{y}^{m}f^{*}(t+i_{1}y+\dots+i_{m}y),$

so that $I_{p,m}^{*}(my)\lesssim I_{p,m}^{*}(y)$. Taking $a_{0}\in X$ as
constructed in Lemma 10 and using Lemmas 11 and 12 with $k=m$, setting
$h=\frac{d}{m}$, we obtain

$\displaystyle\|f-a_{0}\|_{L_{p}(I,X)}^{p}$ $\displaystyle\
\lesssim\frac{1}{d}\int_{0}^{d}I^{\ast}_{p,1}(y)\mathrm{d}y\lesssim\frac{1}{d}\int_{0}^{d}I^{\ast}_{p,m}(y)\mathrm{d}y$
$\displaystyle\
\lesssim\frac{1}{d}\int_{0}^{d}I_{p,m}^{*}\left(\frac{y}{m}\right)\mathrm{d}y=\frac{m}{d}\int_{0}^{\frac{d}{m}}I^{\ast}_{p,m}(y^{\prime})\mathrm{d}y^{\prime}$
$\displaystyle\
\leq\frac{m}{d}\left[2\int_{0}^{\frac{d}{m}}I_{p,m}(y)\mathrm{d}y+c_{p,m}dI_{p,1}\left(\frac{d}{m}\right)\right]$
$\displaystyle\
=c^{\prime}_{p,m}\frac{1}{h}\left[\int_{0}^{h}\int_{a}^{b-my}\|\Delta_{y}^{m}f\|_{X}^{p}\mathrm{d}t\mathrm{d}y+\int_{a}^{b-h}\|\Delta_{h}f(t)\|_{X}^{p}\mathrm{d}t\right],$

which is the desired result. ∎

Finally, a repeated application of Lemma 13 now allows us to establish
Jackson’s inequality.

###### Proof of Theorem 9.

We assume $I=[0,1)$. The general case follows by scaling, using (7). Let
$h=\frac{1}{2r}$, $f\in L_{p}(I,X)$, and denote the approximant $a_{0}\in X$
from Lemma 13 by $M(f,I):=a_{0}$. Now define the coefficients
$a_{0},\ldots,a_{r-1}$ recursively as follows:

$\displaystyle a_{r-1}$
$\displaystyle=M\left(\Delta^{r-1}_{h}f,[0,1-(r-1)h]\right)\frac{1}{h^{r-1}}\frac{1}{(r-1)!},$
$\displaystyle f_{1}(t)$ $\displaystyle=f(t)-a_{r-1}t^{r-1},$ $\displaystyle
a_{r-2}$
$\displaystyle=M(\Delta^{r-2}_{h}f_{1},[0,1-(r-2)h])\frac{1}{h^{r-2}}\frac{1}{(r-2)!},$
$\displaystyle f_{2}(t)$
$\displaystyle=f_{1}(t)-a_{r-2}t^{r-2}=f(t)-(a_{r-1}t^{r-1}+a_{r-2}t^{r-2}),$
$\displaystyle\vdots$ $\displaystyle a_{2}$
$\displaystyle=M(\Delta^{2}_{h}f_{r-3},[0,1-2h])\frac{1}{h^{2}}\frac{1}{2!},$
$\displaystyle f_{r-2}(t)$
$\displaystyle=f_{r-3}(t)-a_{2}t^{2}=f(t)-(a_{r-1}t^{r-1}+a_{r-2}t^{r-2}+\ldots+a_{2}t^{2}),$
$\displaystyle a_{1}$
$\displaystyle=M(\Delta^{1}_{h}f_{r-2},[0,1-h])\frac{1}{h},$ $\displaystyle
f_{r-1}(t)$
$\displaystyle=f_{r-2}(t)-a_{1}t=f(t)-(a_{r-1}t^{r-1}+a_{r-2}t^{r-2}+\ldots+a_{1}t),$
$\displaystyle a_{0}$ $\displaystyle=M(f_{r-1},[0,1]).$

With

$P_{r}(t)=\sum_{k=0}^{r-1}a_{k}t^{k}=a_{0}+a_{1}t+\ldots+a_{r-1}t^{r-1}$

we compute

$\displaystyle\|f-P_{r}\|^{p}_{L_{p}(I,X)}$
$\displaystyle=\|f_{r-1}-a_{0}\|^{p}_{L_{p}(I,X)}=\|f_{r-1}-M(f_{r-1},[0,1])\|^{p}_{L_{p}(I,X)}$
$\displaystyle\lesssim
w_{2r}\left(f_{r-1},I,\frac{1}{2r}\right)_{p}^{p}+\|\Delta_{h}f_{r-1}\|_{L_{p}(I_{h},X)}^{p}$
(which follows from applying Lem. 13 with  $m=2r$)
$\displaystyle=w_{2r}\left(f,I,h\right)_{p}^{p}+\|\Delta_{h}f_{r-2}-a_{1}h\|_{L_{p}(I_{h},X)}^{p}$
$\displaystyle=w_{2r}\left(f,I,h\right)_{p}^{p}+\|\Delta_{h}f_{r-2}-M(\Delta_{h}f_{r-2},[0,1-h])\|_{L_{p}(I_{h},X)}^{p}$
$\displaystyle\lesssim
w_{2r}\left(f,I,h\right)_{p}^{p}+w_{2r-1}\left(\Delta_{h}f_{r-2},[0,1-h],h\right)_{p}^{p}+\|\Delta^{2}_{h}f_{r-2}\|^{p}_{L_{p}(I_{2h},X)}$
(which follows from applying Lem. 13 with $m=2r-1$) $\displaystyle\lesssim
w_{2r-1}\left(f,I,h\right)_{p}^{p}+\|\Delta^{2}_{h}f_{r-3}-a_{2}2!h^{2}\|^{p}_{L_{p}(I_{2h},X)}$
(we used (6))
$\displaystyle=w_{2r-1}\left(f,I,h\right)_{p}^{p}+\|\Delta^{2}_{h}f_{r-3}-M(\Delta^{2}_{h}f_{r-3},[0,1-2h])\|^{p}_{L_{p}(I_{2h},X)}$
$\displaystyle\lesssim
w_{2r-2}\left(f,I,h\right)_{p}^{p}+\|\Delta^{3}_{h}f_{r-4}-a_{3}3!h^{3}\|^{p}_{L_{p}(I_{3h},X)}$
(which follows from applying Lem. 13 with  $m=2r-2$)
$\displaystyle\qquad\vdots$ $\displaystyle\lesssim
w_{r+2}\left(f,I,h\right)_{p}^{p}+\|\Delta^{r-1}_{h}f-a_{r-1}(r-1)!h^{r-1}\|^{p}_{L_{p}(I_{(r-1)h},X)}$
$\displaystyle\lesssim
w_{r+1}\left(f,I,h\right)_{p}^{p}+\|\Delta^{r}_{h}f\|^{p}_{L_{p}(I_{rh},X)}$
(which follows from applying Lem. 13 with  $m=r+1$) $\displaystyle\leq
w_{r+1}\left(f,I,h\right)_{p}^{p}+w_{r}(f,I,h)_{p}^{p}$ $\displaystyle\lesssim
w_{r}\left(f,I,h\right)_{p}^{p},$

which proves the theorem. ∎

######  Remark 15.

Note that Theorem 9 also holds for $p=\infty$: if we extend (25) by

$I^{\ast}_{\infty,k}(h):=\sup_{t\in[a,b)}\|\Delta^{k}_{h}f^{\ast}(t)\|_{X}=\sup_{t\in[0,d)}\|\Delta^{k}_{h}f^{\ast}(t)\|_{X},$

and similarly (26), then Lemmas 10–13 can be extended to the case when
$p=\infty$ by obvious modifications in the proofs, i.e., mostly replacing the
integrals by suprema. In this case the factors ‘$\frac{1}{d}$’ and ‘$d$’ in
Lemmas 10 and 12, respectively, disappear.

###  3.2. Whitney’s estimate

Having established Jackson’s estimate (24) in Theorem 9 we now proceed to
prove Whitney’s estimate.

######  Theorem 16 (Generalized Whitney’s theorem).

Let $0<p,q\leq\infty$, $r\in\mathbb{N}$, and $s>0$. If
$\left(1/q-1/p\right)_{+}\leq s<r$ then there exists a constant $c>0$ which
depends only on $p$, $q$, $r$ such that

$E_{r}(f,I)_{p}=\inf_{P\in\mathbb{V}^{r}_{I,X}}\|f-P\|_{L_{p}(I,X)}\leq
c|I|^{s+\frac{1}{p}-\frac{1}{q}}|f|_{B^{s}_{q,q}(I,X)},$ (36)

for all $f\in B^{s}_{q,q}(I,X)$ and for any finite interval $I$.

Since this involves the $L_{p}$-norm on the left-hand side and an $L_{q}$-norm
on the right-hand side, we first deal with the problem of how to switch from
$p$-norms to $q$-norms for vector-valued polynomials. Using this together with
the Jackson estimate, the fact that according to Lemma 4 we can express the
quasi-norm of the Besov spaces $B^{s}_{p,q}(I,X)$ as a discrete summation
instead of integrals yields Whitney’s estimate.

######  Lemma 17.

Let $0<p<\infty$ and $I=[0,1]$. On $\mathbb{V}^{r}_{I,X}$ the quasi-norm

$\|P\|_{p}:=\left(\int_{0}^{1}\|P(t)\|_{X}^{p}\mathrm{d}t\right)^{1/p}$

is equivalent to the norm

$\|P\|_{\ast}:=\max_{j=1,\ldots,r}\|P_{j}\|_{X},\qquad P_{j}=P(t_{j}),\quad
t_{j}=\frac{j-1}{r-1},\quad j=1,\dots,r.$

The constants involved in the equivalence depend on $r$ and $p$, but are
otherwise independent of $P\in\mathbb{V}_{I,X}^{r}$.

######  Remark 18.

At first sight, it may seem that this lemma is obvious, because it looks like
an equivalence of quasi-norms in a finite-dimensional space. But this is not
the case, since the space $\mathbb{V}^{r}_{I,X}$ is not finite-dimensional,
when $X$ is an arbitrary Banach space.
With slight modifications in the proof, Lemma 17 also holds for $p=\infty$ and
the quasi-norm $\|P\|_{\infty}=\sup_{t\in[0,1]}\|P(t)\|_{X}.$

###### Proof.

Let $\\{\ell_{j}\\}_{j=1}^{r}$ denote the Lagrange basis of $\Pi^{r}$
corresponding to the equally spaced nodes $t_{j}=\frac{j-1}{r-1}$,
$j=1,\dots,r$ on $[0,1]$, i.e.,

$\ell_{j}(t)=\prod_{i\neq j}\frac{t-t_{i}}{t_{j}-t_{i}},\quad\text{so
that}\quad\ell_{j}(t_{i})=\delta_{ij}\quad\text{and}\quad
P=\sum_{j=1}^{r}P_{j}\ell_{j},\text{ if $P\in\mathbb{V}_{I,X}^{r}$}.$

Obviously, for $P\in\mathbb{V}_{I,X}^{r}$,

$\displaystyle\|P\|_{p}$
$\displaystyle=\left(\int_{0}^{1}\|P(t)\|_{X}^{p}\mathrm{d}t\right)^{1/p}=\left(\int_{0}^{1}\Big{\|}\sum_{j=1}^{r}\ell_{j}(t)P_{j}\Big{\|}_{X}^{p}\mathrm{d}t\right)^{1/p}$
$\displaystyle\leq
c_{p,r}\sum_{j=1}^{r}\left(\int_{0}^{1}\ell_{j}^{p}(t)\|P_{j}\|_{X}^{p}\mathrm{d}t\right)^{1/p}\leq
c_{p,r}\max_{j=1,\ldots,r}\|P_{j}\|_{X}=c_{p,r}\|P\|_{\ast}.$

Let now $P=\sum_{j=1}^{r}P_{j}\ell_{j}\in\mathbb{V}_{I,X}^{r}$ and let $i$ be
such that $\|P_{i}\|_{X}=\max_{j}\|P_{j}\|_{X}=\|P\|_{\ast}$. Then, for each
$t\in I$, we have

$\displaystyle\|P(t)\|_{X}$
$\displaystyle=\Big{\|}\sum_{j=1}^{r}\ell_{j}(t)P_{j}\Big{\|}_{X}\geq|\ell_{i}(t)|\|P_{i}\|_{X}-\sum_{j\neq
i}|\ell_{j}(t)|\|P_{j}\|_{X}$
$\displaystyle\geq\|P\|_{\ast}\bigg{(}|\ell_{i}(t)|-\sum_{j\neq
i}|\ell_{j}(t)|\bigg{)}.$ (37)

Since at the point $t=t_{i}$ we have $\ell_{i}(t_{i})=1$ and
$\ell_{j}(t_{i})=0$ for all $j\neq i$, there exists $\delta>0$ such that

$|t-t_{i}|<\delta\quad\Longrightarrow\quad|\ell_{i}(t)|>\frac{3}{4}>\frac{1}{4}>\sum_{j\neq
i}|\ell_{j}(t)|;$

notice that $\delta>0$ can be chosen independent of $i$, but will depend on
$r$. Hence,

$|\ell_{i}(t)|-\sum_{i\neq j}|\ell_{j}(t)|>\frac{1}{2}.$

Hence, (37) gives us

$\|P(t)\|_{X}\geq\frac{1}{2}\|P\|_{\ast}\qquad\text{for}\qquad|t-t_{i}|<\delta.$

Raising to the power $p$ and averaging over the interval
$(t_{i}-\delta,t_{i}+\delta)\cap I$ yields

$\|P\|_{\ast}\leq\left(\frac{2^{p}}{\delta}\int_{(t_{i}-\delta,t_{i}+\delta)\cap
I}\|P(t)\|^{p}_{X}\mathrm{d}t\right)^{1/p}\leq\bar{c}_{p,r}\left(\int_{I}\|P(t)\|^{p}_{X}\mathrm{d}t\right)^{1/p}=\bar{c}_{p,r}\|P\|_{p},$

and the assertion follows. ∎

By a scaling argument we obtain from the previous Lemma the following
equivalence of $L_{p}(I,X)$ norms in $\mathbb{V}_{I,X}^{r}$ on an arbitrary
interval $I$. The proof is very simple and is thus omitted.

######  Corollary 19.

Let $0<p,q\leq\infty$ and $r\in\mathbb{N}$. Then there exists a constant $c>0$
which depends only on $p$, $q$, $r$ such that on any finite interval $I$,

$\|P\|_{L_{p}(I,X)}\leq c|I|^{1/p-1/q}\|P\|_{L_{q}(I,X)},\qquad\forall
P\in\mathbb{V}^{r}_{I,X}.$ (38)

Following the steps from [DeV98, Sec. 6.1] we can now prove Whitney’s estimate
in $B^{s}_{q,q}(I,X)\cap L_{p}(I,X)$.

######  Proposition 20.

Let $0<p,q\leq\infty$, $r\in\mathbb{N}$, and $s>0$. If
$\left(1/q-1/p\right)_{+}\leq s<r$ then there exists a constant $c>0$ which
depends only on $p$, $q$, $r$ such that

$E_{r}(f,I)_{p}:=\inf_{P\in\mathbb{V}^{r}_{I,X}}\|f-P\|_{L_{p}(I,X)}\leq
c|I|^{s+\frac{1}{p}-\frac{1}{q}}|f|_{B^{s}_{q,q}(I,X)},$ (39)

for all $f\in B^{s}_{q,q}(I,X)\cap L_{p}(I,X)$ and for any finite interval
$I$.

###### Proof.

Since $E_{r+1}(f,I)_{p}\leq E_{r}(f,I)_{p}$, it is sufficient to prove the
result in the case $r=\lfloor s\rfloor+1$, and by scaling it is sufficient to
consider $I=[0,1)$. Also, since $E_{r}(f,I)_{p}\leq E_{r}(f,I)_{q}$ when
$p<q$, it is sufficient to consider the case $q\leq p$.

Let $D_{k}$ for $k=0,1,2,\ldots$ denote the following dyadic partitions of
$I$:

$D_{k}:=\\{I_{k}^{j}:=2^{-k}[j-1,j),\ j=1,\ldots,2^{k}\\}.$

We let $S_{k}$ denote a piecewise polynomial function of order $r$ on the
partition $D_{k}$ satisfying the Jackson estimate (24) with $p$ replaced by
$q$, in each sub-interval, i.e.,

$\|f-S_{k}\|_{L_{q}(I^{j}_{k},X)}\lesssim w_{r}(f,I_{k}^{j},2^{-k})_{q},\qquad
j=1,2,\dots,2^{k},\quad k=0,1,\dots,$

whence $S_{0}\in\mathbb{V}_{I,X}^{r}$.

Then, on the one hand, we have

$\displaystyle\|f-S_{k}\|^{q}_{L_{q}(I,X)}$
$\displaystyle=\sum_{j=1}^{2^{k}}\|f-S_{k}\|^{q}_{L_{q}(I^{j}_{k},X)}\lesssim\sum_{j=1}^{2^{k}}w_{r}(f,I_{k}^{j},2^{-k})_{q}^{q}.$

Denoting $\tilde{I}^{j}_{k}=\big{(}I_{k}^{j}\big{)}_{rh}$ we obtain

$\displaystyle\|f-S_{k}\|^{q}_{L_{q}(I,X)}$
$\displaystyle\lesssim\frac{1}{2^{-k}}\int_{0}^{2^{-k}}\sum_{j=1}^{2^{k}}\|\Delta_{h}^{r}f\|^{q}_{L_{q}(\tilde{I}^{j}_{k},X)}\mathrm{d}h$
$\displaystyle=\frac{1}{2^{-k}}\int_{0}^{2^{-k}}\sum_{j=1}^{2^{k}}\int_{\tilde{I}^{j}_{k}}\|\Delta_{h}^{r}f(t)\|_{X}^{q}\mathrm{d}t\mathrm{d}h$
$\displaystyle\leq\frac{1}{2^{-k}}\int_{0}^{2^{-k}}\int_{[0,1-rh]}\|\Delta_{h}^{r}f(t)\|_{X}^{q}\mathrm{d}t\mathrm{d}h$
$\displaystyle=\frac{1}{2^{-k}}\int_{0}^{2^{-k}}\|\Delta_{h}^{r}f\|^{q}_{L_{q}([0,1-rh],X)}\mathrm{d}h=w_{r}(f,I,2^{-k})_{q}^{q}.$
(40)

On the other hand, using (38) in each subinterval $I_{k+1}^{j}$, we have

$\displaystyle\|S_{k}-S_{k+1}\|^{p}_{L_{p}(I,X)}$
$\displaystyle=\sum_{j=1}^{2^{k+1}}\|S_{k}-S_{k+1}\|^{p}_{L_{p}(I_{k+1}^{j},X)}$
$\displaystyle\lesssim
2^{-k\left(1-\frac{p}{q}\right)}\sum_{j=1}^{2^{k+1}}\|S_{k}-S_{k+1}\|^{p}_{L_{q}(I_{k+1}^{j},X)}$
$\displaystyle\lesssim
2^{-k\left(1-\frac{p}{q}\right)}\left(\sum_{j=1}^{2^{k+1}}\|S_{k}-S_{k+1}\|^{q}_{L_{q}(I_{k+1}^{j},X)}\right)^{p/q}$
$\displaystyle=2^{-k\left(1-\frac{p}{q}\right)}\|S_{k}-S_{k+1}\|^{p}_{L_{q}(I,X)},$
(41)

where in the second to last line we used the fact that
$\ell_{q/p}\hookrightarrow\ell_{1}$ for $q\leq p$. This yields for
$\overline{p}=\min\\{1,p\\}$,

$\|S_{k}-S_{k+1}\|^{\overline{p}}_{L_{p}(I,X)}\lesssim
2^{-k\left(\frac{1}{p}-\frac{1}{q}\right)\overline{p}}\|S_{k}-S_{k+1}\|^{\overline{p}}_{L_{q}(I,X)}.$
(42)

But then using (40), (42), and the assumption that $f\in L_{p}(I,X)$, we
obtain

$\displaystyle E_{r}(f,I)_{p}^{\overline{p}}$
$\displaystyle\leq\|f-S_{0}\|^{\overline{p}}_{L_{p}(I,X)}\leq\sum_{k=0}^{\infty}\|S_{k}-S_{k+1}\|^{\overline{p}}_{L_{p}(I,X)}$
$\displaystyle\lesssim\sum_{k=0}^{\infty}2^{-k\left(\frac{1}{p}-\frac{1}{q}\right)\overline{p}}\|S_{k}-S_{k+1}\|^{\overline{p}}_{L_{q}(I,X)}$
$\displaystyle\leq\sum_{k=0}^{\infty}2^{-k\left(\frac{1}{p}-\frac{1}{q}\right)\overline{p}}\left(\|S_{k}-f\|^{\overline{p}}_{L_{q}(I,X)}+\|f-S_{k+1}\|^{\overline{p}}_{L_{q}(I,X)}\right)$
$\displaystyle\lesssim\sum_{k=0}^{\infty}2^{-k\left(\frac{1}{p}-\frac{1}{q}\right)\overline{p}}\|f-S_{k}\|^{\overline{p}}_{L_{q}(I,X)}$
$\displaystyle\lesssim\sum_{k=0}^{\infty}2^{-k\left(\frac{1}{p}-\frac{1}{q}\right)\overline{p}}w_{r}(f,I,2^{-k})_{q}^{\overline{p}}$
$\displaystyle=\sum_{k=0}^{\infty}2^{-k\left(\left(\frac{1}{p}-\frac{1}{q}\right)+s\right)\overline{p}}2^{{ks\overline{p}}}w_{r}(f,I,2^{-k})_{q}^{\overline{p}}$
$\displaystyle\leq\sum_{k=0}^{\infty}2^{-k\delta\overline{p}}2^{{ks\overline{p}}}w_{r}(f,I,2^{-k})_{q}^{\overline{p}},$
(43)

where $\delta:=\left(\frac{1}{p}-\frac{1}{q}\right)+s\geq 0$ due to our
assumption $s\geq\left(1/q-1/p\right)_{+}$. In (43) we proceed as follows: if
$q<\overline{p}$ we make use of the embedding
$\ell_{q}\hookrightarrow\ell_{\overline{p}}$ together with the fact that
$2^{-k\delta\overline{p}}\leq 1$ and for $q>\overline{p}$ we apply Hölder’s
inequality with $\frac{q}{\overline{p}}>1$. This finally gives

$\|f-S_{0}\|_{L_{p}(I,X)}\leq\left(\sum_{k=0}^{\infty}2^{{ksq}}w_{r}(f,2^{-k},I)_{q}^{q}\right)^{1/{q}}\simeq|f|_{B^{s}_{q,q}(I,X)}.$
(44)

The assertion thus follows by recalling that $S_{0}\in\mathbb{V}_{I,X}^{r}$. ∎

As a consequence of the previous theorem we have that under the same
assumptions $B^{s}_{q,q}(I,X)$ is embedded into $L_{p}(I,X)$.

######  Corollary 21.

Let $0<p,q\leq\infty$, $r\in\mathbb{N}$, and $s>0$. If
$\left(1/q-1/p\right)_{+}\leq s$ then $B^{s}_{q,q}(I,X)$ is embedded into
$L_{p}(I,X)$ and there exists a constant $c>0$ which depends only on $p$, $q$,
$r$, and $s$ such that

$\|f\|_{L_{p}(I,X)}\leq c\|f\|_{B^{s}_{q,q}(I,X)},$

for all $f\in B^{s}_{q,q}(I,X)$ and for any finite interval $I$.

###### Proof.

Let $f\in B^{s}_{q,q}(I,X)\cap L_{p}(I,X)$, $r\in\mathbb{N}$, $r>s$, and let
$S_{0}$ be as in the proof of Theorem 16. Then,

$\|f\|_{L_{p}(I,X)}\lesssim\|f-S_{0}\|_{L_{p}(I,X)}+\|S_{0}\|_{L_{p}(I,X)}\lesssim|f|_{B^{s}_{q,q}(I,X)}+\|S_{0}\|_{L_{q}(I,X)}.$

Since $S_{0}\in\mathbb{V}_{I,X}^{r}$ was chosen satisfying Jackson estimate
(24) with $p$ replaced by $q$,

$\|S_{0}\|_{L_{q}(I,X)}\lesssim\|f-S_{0}\|_{L_{q}(I,X)}+\|f\|_{L_{q}(I,X)}\lesssim
w_{r}(f,I,1)_{q}+\|f\|_{L_{q}(I,X)}\lesssim\|f\|_{L_{q}(I,X)}.$

Therefore, for all $f\in B^{s}_{q,q}(I,X)\cap L_{p}(I,X)$,

$\|f\|_{L_{p}(I,X)}\lesssim|f|_{B^{s}_{q,q}(I,X)}+\|f\|_{L_{q}(I,X)}\lesssim\|f\|_{B^{s}_{q,q}(I,X)}.$

Finally, since $B^{s}_{q,q}(I,X)\cap L_{p}(I,X)$ is dense in
$B^{s}_{q,q}(I,X)$ the assertion follows. ∎

The generalized Whitney’s theorem, Theorem 16, is now a consequence of
Proposition 20 and Corollary 21.

##  4\. Adaptive approximation in one variable

###  4.1. The stationary case

Given a polyhedral space domain $\Omega\subset{}^{n}$, $n\in\mathbb{N}$, we
let $\mathbb{T}(\mathcal{T}_{0})$ denote the set of all triangulations
$\mathcal{T}$ (partitions into simplices) that are obtained by successive
application of the bisection routine of [Ste08] from a properly labeled
initial triangulation $\mathcal{T}_{0}$ of $\Omega$. If $n=1$,
$\mathbb{T}(\\{0<T\\})$ denotes the set of all partitions of $\Omega=[0,T)$
into sub-intervals that may be obtained by successive bisection of
$\mathcal{T}_{0}=\\{[0,T)\\}$. For simplicity, the one-dimensional partition
$\\{[0=t_{0},t_{1}),[t_{1},t_{2}),\dots,[t_{N-1},t_{N}=T)\\}$ will be usually
denoted by $\\{0=t_{0}<t_{1}<\dots<t_{N}=T\\}$. Whenever we write
$\mathcal{T}_{*}=\textsc{Refine($\mathcal{T},\mathcal{M}$)}$, we understand
that $\mathcal{M}\subset\mathcal{T}$ and $\mathcal{T}_{*}$ is the refinement
of $\mathcal{T}$ obtained by the bisection routine of [Ste08]. In the one-
dimensional case, we understand that $\mathcal{T}_{*}$ is obtained by the sole
replacement in $\mathcal{T}$ of each element $T=[a,b)\in\mathcal{M}$ by its
children $[a,\frac{a+b}{2})$, $[\frac{a+b}{2},b)$.

Therefore, the following complexity bound holds:

> Let $\mathcal{T}_{0}$, $\mathcal{T}_{1}$, $\mathcal{T}_{2}$, …, be a
> sequence of partitions in $\mathbb{T}(\mathcal{T}_{0})$ obtained by
> successive calls of
> $\mathcal{T}_{k+1}=\textsc{Refine($\mathcal{T}_{k},\mathcal{M}_{k}$)}$, with
> $\mathcal{M}_{k}\subset\mathcal{T}_{k}$ the set of _marked_ elements. Then,
> there exists a constant $C$ that depends on the initial triangulation
> $\mathcal{T}_{0}$ such that
>
> $\\#\mathcal{T}_{k}-\\#\mathcal{T}_{0}\leq
> C\sum_{j=0}^{k-1}\\#\mathcal{M}_{j},\qquad k=1,2,\dots.$ (45)

For $\mathcal{T}\in\mathbb{T}(\mathcal{T}_{0})$, recall that
$\mathbb{V}_{\mathcal{T}}^{r}$ is the finite element space of continuous
piecewise polynomials of order $r$, i.e.,

$\mathbb{V}_{\mathcal{T}}^{r}:=\\{g\in C(\overline{\Omega}):\
g\big{|}_{T}\in\Pi^{r}\ \text{for all }T\in\mathcal{T}\\},$

where $\Pi^{r}$ denotes the set of polynomials of total degree (strictly) less
than $r$. The underlying domain $\Omega$ and its dimension are implicitly
indicated by the partition $\mathcal{T}$, which will sometimes correspond to a
time interval $[0,T)$ and sometimes to an $n$-dimensional space domain.

##### Approximation Classes

Let $X$ be a quasi-Banach space on the polyhedral bounded Lipschitz domain
$\Omega\subset\mathbb{R}^{n}$ with quasi-norm $\|\cdot\|_{X}$. Let
$\mathcal{T}_{0}$ be a triangulation of $\Omega$, properly labeled so that
(45) holds, and assume further that $\mathbb{V}_{\mathcal{T}}^{r}\subset X$
for $\mathcal{T}\in\mathbb{T}(\mathcal{T}_{0})$. In this context, for $f\in
X$, the best $N$-term approximation error is given by

$\sigma_{N}(f)=\inf_{|\mathcal{T}|\leq
N}\inf_{g\in\mathbb{V}_{\mathcal{T}}^{r}}\|f-g\|_{X}.$

For $s>0$ we define the approximation class $\mathbb{A}_{s}(X)$ as the set of
those functions in $X$ whose best $N$-term approximation error is of order
$N^{-s}$, i.e.,

$\mathbb{A}_{s}(X):=\\{f\in X:\ \exists c>0\text{ such that }\sigma_{N}(f)\leq
cN^{-s},\ \forall N\in\mathbb{N}\\}.$

Equivalently, we can define $\mathbb{A}_{s}(X)$ through a semi-quasi-norm as
follows:

$\mathbb{A}_{s}(X):=\\{f\in X:\
|f|_{\mathbb{A}_{s}(X)}<\infty\\}\quad\text{with}\quad|f|_{\mathbb{A}_{s}(X)}:=\sup_{N\in\mathbb{N}}N^{s}\sigma_{N}(f).$

Alternatively, this definition is equivalent to saying that
$f\in{\mathbb{A}_{s}(X)}$ if there is a constant $c$ such that for all
$\varepsilon>0$, there exists a mesh $\mathcal{T}$ that satisfies

$\inf_{g\in\mathbb{V}^{r}_{\mathcal{T}}}\|f-g\|_{X}\leq
c\varepsilon\quad\text{and}\quad|\mathcal{T}|\leq\varepsilon^{-1/s},$ (46)

and $|f|_{\mathbb{A}_{s}(X)}$ is equivalent to the infimum of all constants
$c$ that satisfy (46).

We use the following result from [GM14, Thm. 2.2, Cor. 2.3], which is the
high-order analog to the one presented in [BDDP02] for linear finite elements
($r=2$).

######  Theorem 22.

Let $X=B^{\alpha}_{p,p}(\Omega)$, $0<p<\infty$,
$0<\alpha<\min\\{r,1+\frac{1}{p}\\}$ or $X=L_{p}(\Omega)$ if $\alpha=0$. If
$f\in B^{s+\alpha}_{\tau,\tau}(\Omega)$ with $s>0$,
$0<\frac{1}{\tau}<\frac{s}{n}+\frac{1}{p}$, and $s+\alpha<r$, then

$\displaystyle B^{\alpha+s}_{\tau,\tau}(\Omega)$
$\displaystyle\subset\mathbb{A}_{s/n}(B^{\alpha}_{p,p}(\Omega))\qquad(\alpha>0),$
(47) $\displaystyle B^{s}_{\tau,\tau}(\Omega)$
$\displaystyle\subset\mathbb{A}_{s/n}(L_{p}(\Omega))\qquad\ \;(\alpha=0).$
(48)

In particular, if $p=2$ and $\alpha=0$ we have the following result.

######  Corollary 23.

Let $X=L_{2}(\Omega)$, $r\in\mathbb{N}$, $0<s<r$, and
$0<\frac{1}{\tau}<\frac{s}{n}+\frac{1}{2}$. Then there exists a constant
$C=C(r,s,\tau,\Omega,\mathbb{T})$ such that, for every $\varepsilon>0$ there
exists $\mathcal{T}\in\mathbb{T}(\mathcal{T}_{0})$ and
$g\in\mathbb{V}^{r}_{\mathcal{T}}$ such that

$\|f-g\|_{X}\leq\varepsilon\,|f|_{B^{s}_{\tau,\tau}(\Omega)}\qquad\text{and}\qquad|\mathcal{T}|\lesssim\varepsilon^{-n/s}.$

###  4.2. Greedy algorithm

Theorem 22, or equivalently Corollary 23, is proved with the help of a so
called _Greedy algorithm_. In order to make this article self-contained, we
present it here and use it to build a quasi-optimal partition of $[0,T)$ to
approximate a vector-valued function in $L_{p}([0,T),X)$. This, in turn, is an
intermediate tool for constructing the optimal time-space partition.

In the rest of this section we consider the following framework. We let $X$
denote a Banach space, $r\in\mathbb{N}$ denotes the polynomial _order_ with
respect to time, and for an interval $I$, recall the definition of
$\mathbb{V}_{I,X}^{r}$ from (21):

$\mathbb{V}^{r}_{I,X}:=\Big{\\{}P(t)=\sum_{j=0}^{r-1}a_{j}t^{j},\ a_{j}\in X,\
t\in I\Big{\\}}\subset L_{p}(I,X),$

i.e., the tensor product space $\Pi^{r}\otimes X$ on the time slice
$I\times\Omega$. For a partition
$\mathcal{T}=\\{0=t_{0}<t_{1}<\dots<t_{N}=T\\}$ of the time interval $[0,T)$,
we consider the following corresponding (abstract) finite element space:

$\mathbb{V}_{\mathcal{T},X}^{r}=\\{P\in L_{p}([0,T),X):\
P_{|I}\in\mathbb{V}_{I,X}^{r},\ I\in\mathcal{T}\\}.$

Recall the definition of the best approximation error $E_{r}(f,I)_{p}$
associated with an interval $I\subset[0,T)$, i.e.,

$E_{r}(f,I)_{p}=\inf_{P_{I}\in\mathbb{V}_{I,X}^{r}}\|f-P_{I}\|_{L_{p}(I,X)},$
(49)

so that

$\inf_{g\in\mathbb{V}_{\mathcal{T},X}^{r}}\|f-g\|_{L_{p}([0,T),X)}=\left(\sum_{I\in\mathcal{T}}E_{r}(f,I)_{p}^{p}\right)^{1/p}.$

An algorithm approximating the solution with a parameter $\delta>0$ reads as
follows:

Algorithm 1 _Greedy_ algorithm

1:function Greedy($f$,$\delta$)

2: Let $\mathcal{T}_{0}=\\{0<T\\}=\\{[0,T)\\}$.

3: $k=0$

4: while
$\mathcal{M}_{k}:=\\{I\in\mathcal{T}_{k}:E_{r}(f,I)_{p}>\delta\\}\neq\emptyset$
do

5: Let $\mathcal{T}_{k+1}=\textsc{Refine}(\mathcal{T}_{k},\mathcal{M}_{k})$

6: $k\leftarrow k+1$

7: end while

8:end function

###  4.3. Semi-discretization in time

Concerning the error when approximating a vector-valued function with
piecewise polynomials with respect to time, we have the following result.

######  Theorem 24 (Time discretization).

Let $X$ be a separable Banach space, let $s>0$, $0<p,q\leq\infty$, and
$\left(\frac{1}{q}-\frac{1}{p}\right)_{+}\leq s<r$, with $r\in\mathbb{N}$.
Then, if $f\in B^{s}_{q,q}([0,T),X)$ and $\varepsilon>0$, there exists
$\delta>0$ such that Greedy($f$,$\delta$) terminates in finitely many steps
and the generated partition $\mathcal{T}$ satisfies

$\\#\mathcal{T}\leq c_{1}\,\varepsilon^{-1/s},$ (50)

where the constant $c_{1}>0$ depends on $p$, $q$, and $s$ but not on $f$.
Moreover, there exists $P\in\mathbb{V}_{\mathcal{T},X}^{r}$ satisfying

$\|f-P\|_{L_{p}([0,T),X)}\leq
c_{2}\,\varepsilon\,|f|_{B^{s}_{q,q}([0,T),X)}\leq
c_{3}\,{(\\#\mathcal{T})^{-s}}|f|_{B^{s}_{q,q}([0,T),X)},$ (51)

with $c_{2},c_{3}>0$ depending on $p$, $q$, and $s$ but not on $f$.

###### Proof.

Let $\varepsilon>0$ be given and let
$\delta=\varepsilon^{\frac{s+1/p}{s}}|f|_{B^{s}_{q,q}([0,T),X)}$. Using
Whitney’s estimate (36) we see that the error $E_{r}(f,I)_{p}$ associated with
an interval $I$ satisfies

$E_{r}(f,I)_{p}=\inf_{P_{I}\in\mathbb{V}_{I,X}^{r}}\|f-P_{I}\|_{L_{p}(I,X)}\lesssim|I|^{s+\frac{1}{p}-\frac{1}{q}}|f|_{B^{s}_{q,q}(I,X)}.$
(52)

Since $s+\frac{1}{p}-\frac{1}{q}>0$ the right-hand side goes to zero as $|I|$
goes to zero, which shows that the Greedy algorithm terminates in a finite
number of steps $K$.

We now bound the number of elements of $\mathcal{T}:=\mathcal{T}_{K}$ as
follows. Initially, $\mathcal{T}_{0}=\\{[0,T)\\}$, therefore,
$\\#\mathcal{T}_{0}=1$. In each iteration of the while-loop,
$\\#\mathcal{M}_{k}$ elements are _marked_ for refinement. If
$\overline{\mathcal{M}}=\bigcup_{k=0}^{K-1}\mathcal{M}_{k}$ is the union of
all marked elements in a certain step of the algorithm, then, due to (45), the
resulting final partition $\mathcal{T}$ satisfies $\\#\mathcal{T}\lesssim
1+\\#\overline{\mathcal{M}}\lesssim\\#\overline{\mathcal{M}}$. We see that
estimating $\\#\mathcal{T}$ is comparable with estimating
$\\#\overline{\mathcal{M}}$. In order to count the number of elements in
$\overline{\mathcal{M}}$ observe that
$\\#\overline{\mathcal{M}}=\sum_{k=0}^{\infty}\\#\mathcal{M}^{k}$, with

$\mathcal{M}_{k}=\left\\{I\in\overline{\mathcal{M}}:|I|=T/2^{k}\right\\}\
\text{ if }\ 0\leq k\leq K-1\qquad\text{and}\qquad\mathcal{M}_{k}=\emptyset\
\text{ if }\ k\geq K.$

On the one hand, since our time interval $[0,T)$ is finite, we obtain the
upper bound

$\\#\mathcal{M}_{k}\leq 2^{k},\qquad k\in\mathbb{N}_{0}.$

On the other hand, if $I\in\mathcal{M}_{k}$ from steps 4 and 6 of the Greedy
algorithm and formula (52), we have

$\delta<E_{r}(f,I)_{p}\lesssim\left(\frac{1}{2^{k}}\right)^{s+\frac{1}{p}-\frac{1}{q}}|f|_{B^{s}_{q,q}(I,X)},\quad\text{so
that}\quad\delta^{q}\lesssim\left(\frac{1}{2^{k}}\right)^{sq+\frac{q}{p}-1}|f|^{q}_{B^{s}_{q,q}(I,X)}.$

This implies

$\delta^{q}\\#\mathcal{M}_{k}=\sum_{I\in\mathcal{M}_{k}}\delta^{q}\lesssim\left(\frac{1}{2^{k}}\right)^{sq+\frac{q}{p}-1}\sum_{I\in\mathcal{M}_{k}}|f|^{q}_{B^{s}_{q,q}(I,X)}\leq\left(\frac{1}{2^{k}}\right)^{sq+\frac{q}{p}-1}|f|^{q}_{B^{s}_{q,q}([0,T),X)},$

i.e.,

$\\#\mathcal{M}_{k}\lesssim\min\left\\{2^{k},\frac{1}{\delta^{q}}\left(\frac{1}{2^{k}}\right)^{sq+\frac{q}{p}-1}|f|^{q}_{B^{s}_{q,q}([0,T),X)}\right\\}.$

The first term corresponds to an increasing geometric series, the second to a
decreasing one. Setting $k_{0}:=\min\left\\{k\in\mathbb{N}_{0}:\
\frac{1}{\delta^{q}}\left(\frac{1}{2^{k}}\right)^{sq+\frac{q}{p}-1}|f|^{q}_{B^{s}_{q,q}([0,T),X)}<2^{k}\right\\}$
we obtain

$\displaystyle\\#\overline{\mathcal{M}}=\sum_{k=0}^{\infty}\\#\mathcal{M}_{k}$
$\displaystyle\leq\sum_{k=0}^{k_{0}-1}2^{k}+\sum_{k=k_{0}}^{\infty}\frac{1}{\delta^{q}}\left(\frac{1}{2^{k}}\right)^{sq+\frac{q}{p}-1}|f|^{q}_{B^{s}_{q,q}([0,T),X)}$
$\displaystyle\lesssim
2^{k_{0}}+\frac{1}{\delta^{q}}\left(\frac{1}{2^{k_{0}}}\right)^{sq+\frac{q}{p}-1}|f|^{q}_{B^{s}_{q,q}([0,T),X)}\lesssim
2^{k_{0}}.$ (53)

In order to estimate $2^{k_{0}}$ we observe that

$2^{k_{0}-1}\leq\frac{1}{\delta^{q}}\left(\frac{1}{2^{k_{0}}}\right)^{sq+\frac{q}{p}-1}|f|^{q}_{B^{s}_{q,q}([0,T),X)}<2^{k_{0}},$
$2^{k_{0}(s+\frac{1}{p})-\frac{1}{q}}\leq\frac{1}{\delta}|f|_{B^{s}_{q,q}([0,T),X)}<2^{k_{0}(s+\frac{1}{p})}.$

We see that

$2^{k_{0}}\leq\left(\frac{1}{\delta}\right)^{\frac{1}{s+1/p}}|f|^{\frac{1}{s+1/p}}_{B^{s}_{q,q}([0,T),X)},$
(54)

therefore, from (53) and (54) we get

$\\#\mathcal{T}\lesssim\\#\overline{\mathcal{M}}\lesssim\left(\frac{1}{\delta}\right)^{\frac{1}{s+1/p}}|f|^{\frac{1}{s+1/p}}_{B^{s}_{q,q}([0,T),X)},\quad\text{i.e.},\quad\delta\lesssim(\\#\mathcal{T})^{-(s+\frac{1}{p})}|f|_{B^{s}_{q,q}([0,T),X)},$

and (50) follows after recalling that
$\delta=\varepsilon^{\frac{s+1/p}{s}}|f|_{B^{s}_{q,q}([0,T),X)}$.

Finally, for each $I\in\mathcal{T}$ we let $P_{I}\in\mathbb{V}_{I,X}^{r}$
satisfy $\|f-P_{I}\|_{L_{p}(I,X)}\leq{2E_{r}(f,I)_{p}}$ and let
$P(t)=\sum_{I\in\mathcal{T}}\chi_{I}(t)P_{I}(t)$, $t\in[0,T)$. Hence,

$\displaystyle
P\in\mathbb{V}_{\mathcal{T},X}^{r}\quad\text{and}\quad\|f-P\|_{L_{p}([0,T),X)}^{p}$
$\displaystyle\lesssim\delta^{p}\\#\mathcal{T}\lesssim\\#\mathcal{T}^{-ps}|f|_{B^{s}_{q,q}([0,T),X)}^{p},$

and (51) follows. ∎

##  5\. Discretization in time and space

We now consider the error when approximating a function with piecewise
polynomials with respect to time and space. In this article, we deal with the
approximation in $L_{2}([0,T)\times\Omega)=L_{2}([0,T),X)$, where hereafter we
let $X=L_{2}(\Omega)$. We restrict ourselves to this Hilbertian case in order
to avoid additional technical difficulties and leave the study of more general
quasi-norms, e.g. $p\neq 2$ and $X\neq L_{2}(\Omega)$, to a forthcoming
article.

###  5.1. Time marching fully discrete adaptivity

Recall that the type of discretizations that we consider are those consisting
of a partition $\\{0=t_{0}<t_{1}<\dots<t_{N}=T\\}$ of the time interval and a
sequence of partitions $\mathcal{T}_{1},\dots,\mathcal{T}_{N}\in\mathbb{T}$ of
the space domain $\Omega$, where $\mathcal{T}_{i}$ corresponds to the
subinterval $[t_{i-1},t_{i})$, $i=1,\dots N$. The time-space partition is then
given by

$\mathcal{P}=\left(\\{0=t_{0}<t_{1}<\ldots<t_{N}=T\\},\\{\mathcal{T}_{1},\ldots,\mathcal{T}_{N}\\}\right),\quad\text{with
}\\#\mathcal{P}=\sum_{i=1}^{N}\\#\mathcal{T}_{i},$

Given $r_{1},r_{2}\in\mathbb{N}$, the finite element space
$\overline{\mathbb{V}}_{\mathcal{P}}^{r_{1},r_{2}}$ subject to such a
partition $\mathcal{P}$ is defined as

$\displaystyle\overline{\mathbb{V}}_{\mathcal{P}}^{r_{1},r_{2}}$
$\displaystyle:=\\{G:[0,T)\times\Omega\rightarrow\real:G_{\big{|}[t_{i-1},t_{i})\times\Omega}\in\Pi^{r_{1}}\otimes\mathbb{V}_{\mathcal{T}_{i}}^{r_{2}},\text{
for all }i=1,2,\dots,N\\},$

i.e., $G\in\overline{\mathbb{V}}_{\mathcal{P}}^{r_{1},r_{2}}$ if and only if
$G(t,\cdot)\in\mathbb{V}_{\mathcal{T}_{i}}^{r_{2}}$ for all
$t\in[t_{i-1},t_{i})$ and $G(\cdot,x)\big{|}_{[t_{i-1},t_{i})}\in\Pi^{r_{1}}$
for all $x\in\Omega$ and all $i=1,2,\dots,N$.

In order to construct an optimal approximate solution with tolerance
$\varepsilon>0$ we use the one-dimensional Greedy algorithm as described on
page 1 for the (adaptive) discretization in time and an $n$-dimensional Greedy
algorithm for (adaptive) discretizations in space. This allows us to use the
results from Theorems 24 and 22, respectively. In particular, we obtain the
following result.

######  Theorem 25 (Approximation with fully discrete functions).

Let $0<s_{i}<r_{i}$, $i=1,2$, $0<q_{1}\leq\infty$, $1\leq q_{2}\leq\infty$
with $s_{1}>\big{(}\frac{1}{q_{1}}-\frac{1}{2}\big{)}_{+}$ and
$s_{2}>n(\frac{1}{q_{2}}-\frac{1}{2}\big{)}_{+}$. Let $f\in
B^{s_{1}}_{q_{1},q_{1}}([0,T),X)\cap
L_{2}([0,T),B_{q_{2},q_{2}}^{s_{2}}(\Omega))$, with $X=L_{2}(\Omega)$. Then,
for each $\varepsilon>0$ there exists a time-space partition $\mathcal{P}$
that satisfies

$\\#\mathcal{P}\leq
c_{1}\varepsilon^{-\big{(}\frac{1}{s_{1}}+\frac{n}{s_{2}}\big{)}}$

and a function $F\in\overline{\mathbb{V}}_{\mathcal{P}}^{r_{1},r_{2}}$ such
that

$\displaystyle\|f-F\|_{L_{2}([0,T),X)}\leq
c_{2}\varepsilon|\\!|\\!|f|\\!|\\!|\leq
c_{3}(\\#\mathcal{P})^{-\frac{1}{\frac{1}{s_{1}}+\frac{n}{s_{2}}}}|\\!|\\!|f|\\!|\\!|,$

where
$|\\!|\\!|f|\\!|\\!|=|f|_{B^{s_{1}}_{q_{1},q_{1}}([0,T),X)}+\|f\|_{L_{2}([0,T),B_{q_{2},q_{2}}^{s_{2}}(\Omega))}$
and the positive constants $c_{1},c_{2},c_{3}$ depend on $q_{1},q_{2}$, and
$s_{1},s_{2}$ but not on $f$.

######  Remark 26.

Here (with a little abuse) we use the notation

$L_{2}(I,B_{q,q}^{s}(\Omega))=\left\\{f:I\to
B_{q,q}^{s}(\Omega):\|f\|_{L_{2}(I,B_{q,q}^{s}(\Omega))}<\infty\right\\}$

with
$\|f\|_{L_{2}(I,B_{q,q}^{s}(\Omega))}:=\left(\int_{I}\|f(t)\|_{B_{q,q}^{s}(\Omega)}^{2}\mathrm{d}t\right)^{1/2}$.

The restriction $q_{2}\geq 1$ in Theorem 25 can probably be removed and
replaced by $q_{2}>0$. It appears here due to the fact that we require in the
proof below a uniform bound of the approximants on a subinterval
$I=[t_{i-1},t_{i})$, which is established in Lemma 27. Our current proof of
Lemma 27 uses Minkowski’s inequality which only works if $q_{2}\geq 1$. So far
we were not able to find an appropriate modification for $q_{2}<1$.

###### Proof of Theorem 25.

Given $f\in B^{s_{1}}_{q_{1},q_{1}}([0,T),X)\cap
L_{2}([0,T),B_{q_{2},q_{2}}^{s_{2}}(\Omega))$ and $\varepsilon>0$, the
approximant $F\in\overline{\mathbb{V}}_{\mathcal{P}}^{r_{1},r_{2}}$ is
constructed in two steps as follows.

We first use a one-dimensional Greedy algorithm and apply the results from
Theorem 24. This gives a partition of the time interval
$0=t_{0}<t_{1}<\ldots<t_{N}=T$ and an approximant
$G=\sum_{i=1}^{N}{\chi}_{[t_{i-1},t_{i})}G_{i}\in\mathbb{V}_{\\{0<t_{1}<\dots<T\\},X}^{r_{1}}$
with $G_{i}$ the $L_{2}([t_{i-1},t_{i}),X)$ projection of
$f_{|[t_{i-1},t_{i})}$ into $\mathbb{V}_{[t_{i-1},t_{i}),X}^{r_{1}}$. This
partition and approximant satisfy

$N\lesssim\varepsilon^{-1/s_{1}}\quad\text{and}\quad\|f-G\|_{L_{2}([0,T),X)}\lesssim\varepsilon|f|_{B^{s_{1}}_{q_{1},q_{1}}([0,T),X)}.$

Also, if $\\{W_{i}^{j}\\}_{j=1}^{r_{1}}$ is an orthonormal basis of
$\mathbb{V}^{r_{1}}_{[t_{i-1},t_{i}),\mathbb{R}}$ then

$G_{i}(t)=\sum_{j=1}^{r_{1}}G_{i}^{j}\,W_{i}^{j}(t),\quad\text{with}\quad
G_{i}^{j}=\int_{I}f(t)W_{i}^{j}(t)\,\mathrm{d}t,$

noting that the last integral is a Bochner integral in $X=L_{2}(\Omega)$.

We now observe that due to Lemma 27 below we have $G_{i}^{j}\in
B^{s_{2}}_{q_{2},q_{2}}(\Omega)$ and

$\|G_{i}\|_{L_{2}([t_{i-1},t_{i}),B^{s_{2}}_{q_{2},q_{2}}(\Omega))}\lesssim\|f\|_{L_{2}([t_{i-1},t_{i}),B^{s_{2}}_{q_{2},q_{2}}(\Omega))}.$
(55)

The second step consists in approximating each function $G_{i}^{j}\in
B^{s_{2}}_{q_{2},q_{2}}(\Omega)$ using the space-adaptive Greedy algorithm.
Resorting to Corollary 23 we find a mesh
$\mathcal{T}_{i}^{j}\in\mathbb{T}(\mathcal{T}_{0})$ and a finite element
function $F_{i}^{j}\in\mathbb{V}_{\mathcal{T}_{i}^{j}}^{r_{2}}$ with

$\\#\mathcal{T}_{i}^{j}\lesssim\varepsilon^{-\frac{n}{s_{2}}}\quad\text{and}\quad\|G_{i}^{j}-F_{i}^{j}\|_{X}\lesssim\varepsilon|G_{i}^{j}|_{B^{s_{2}}_{q_{2},q_{2}}(\Omega)}.$

Therefore, after defining
$\mathcal{T}_{i}=\oplus_{j=1}^{r_{1}}\mathcal{T}_{i}^{j}$ (the overlay of the
meshes [CKNS08]), we have that
$F_{i}(t):=\sum_{j=1}^{r_{1}}W_{i}^{j}(t)F_{i}^{j}\in\mathbb{V}_{[t_{i-1},t_{i}),\mathbb{V}_{\mathcal{T}_{i}}^{r_{2}}}^{r_{1}}$
satisfies

$\displaystyle\\#\mathcal{T}_{i}\leq\sum_{j=1}^{r_{1}}\\#\mathcal{T}_{i}^{j}$
$\displaystyle\lesssim\varepsilon^{-\frac{n}{s_{2}}}\quad\text{\cite[cite]{[\@@bibref{}{CKNS08}{}{},
Lem.~{}3.7]} and}\quad$
$\displaystyle\|F_{i}-G_{i}\|_{L_{2}([t_{i-1},t_{i}),X)}$
$\displaystyle\lesssim\varepsilon\|G_{i}\|_{L_{2}([t_{i-1},t_{i}),B^{s_{2}}_{q_{2},q_{2}}(\Omega))}\lesssim\varepsilon\|f\|_{L_{2}([t_{i-1},t_{i}),B^{s_{2}}_{q_{2},q_{2}}(\Omega))},$

due to (55).

Finally, we let
$\mathcal{P}=\\{\\{0=t_{0}<t_{1}<\dots<t_{N}=T\\},\\{\mathcal{T}_{1},\mathcal{T}_{2},\dots,\mathcal{T}_{N}\\}\\}$
and define
$F=\sum_{i=1}^{N}\chi_{[t_{i-1},t_{i})}F_{i}\in\overline{\mathbb{V}}_{\mathcal{P}}^{r_{1},r_{2}}$,
whence by the triangle inequality

$\displaystyle\|f-F\|_{L_{2}([0,T),X)}$
$\displaystyle\leq\|f-G\|_{L_{2}([0,T),X)}+\|G-F\|_{L_{2}([0,T),X)}$
$\displaystyle\lesssim\varepsilon|f|_{B^{s_{1}}_{q_{1},q_{1}}([0,T),X)}+\bigg{(}\sum_{i=1}^{N}\|G_{i}-F_{i}\|_{L_{2}([t_{i-1},t_{i}),X)}^{2}\bigg{)}^{1/2}$
$\displaystyle\lesssim\varepsilon\Big{(}|f|_{B^{s_{1}}_{q_{1},q_{1}}([0,T),X)}+\|f\|_{L_{2}([t_{i-1},t_{i}),B^{s_{2}}_{q_{2},q_{2}}(\Omega))}\Big{)}$

and

$\displaystyle\\#\mathcal{P}=\sum_{i=1}^{N}\\#\mathcal{T}_{i}\lesssim
N\,\varepsilon^{-\frac{n}{s_{2}}}\lesssim\varepsilon^{-\frac{1}{s_{1}}}\,\varepsilon^{-\frac{n}{s_{2}}}=\varepsilon^{-(\frac{1}{s_{1}}+\frac{n}{s_{2}})}.$

The assertion of the theorem thus follows. ∎

In Theorem 25, formula (55), we required a uniform bound of the approximants
$G_{i}$ on a subinterval $I=[t_{i-1},t_{i})$, which is provided by the
following lemma.

######  Lemma 27.

Given a finite interval $I$, let $r=r_{1}$, $s=s_{2}$, and $q=q_{2}$ satisfy
the assumptions from Theorem 25 and assume $f\in
L_{2}(I,B_{q,q}^{s}(\Omega))$. If $G\in\mathbb{V}_{I,X}^{r}$ is the
$L_{2}(I,X)$ projection of $f\in L_{2}(I,B_{q,q}^{s}(\Omega))$, then

$\|G\|_{L_{2}(I,B^{s}_{q,q}(\Omega))}\lesssim\|f\|_{L_{2}(I,B^{s}_{q,q}(\Omega))}.$

###### Proof.

If $\\{W^{j}\\}_{j=1}^{r}$ is an orthonormal basis of
$\mathbb{V}_{I,\mathbb{R}}^{r}$ then

$G(t)=\sum_{j=1}^{r}G^{j}W^{j}(t)\quad\text{with}\quad
G^{j}=\int_{I}f(t)W^{j}(t)\mathrm{d}t,$

i.e.,

$G(t)(x)=\sum_{j=1}^{r}G^{j}(x)W^{j}(t)\quad\text{with}\quad
G^{j}(x)=\int_{I}f(t,x)W^{j}(t)\mathrm{d}t,$

for almost every $x\in\Omega$. Notice first that

$\displaystyle\|G\|_{L_{2}(I,B^{s}_{q,q}(\Omega))}^{2}$
$\displaystyle=\int_{I}\|G(t)\|_{L_{q}(\Omega)}^{2}+|G|_{B_{q,q}^{s}(\Omega)}^{2}\mathrm{d}t$
$\displaystyle=\int_{I}\Big{\|}\sum_{j=1}^{r}G^{j}W^{j}(t)\Big{\|}_{L_{q}(\Omega)}^{2}+\Big{|}\sum_{j=1}^{r}G^{j}W^{j}(t)\Big{|}_{B_{q,q}^{s}(\Omega)}^{2}\mathrm{d}t$
$\displaystyle\lesssim\sum_{j=1}^{r}\|G^{j}\|_{L_{q}(\Omega)}^{2}+|G^{j}|_{B_{q,q}^{s}(\Omega)}^{2},$

so that

$\|G\|_{L_{2}(I,B^{s}_{q,q}(\Omega))}\lesssim\sum_{j=1}^{r}\|G^{j}\|_{L_{q}(\Omega)}+|G^{j}|_{B_{q,q}^{s}(\Omega)}.$
(56)

We now bound $\|G^{j}\|_{L_{q}(\Omega)}$ and $|G^{j}|_{B_{q,q}^{s}(\Omega)}$
and focus on the case $1\leq q<\infty$, noting that the case $q=\infty$ is
analogous. Since $q\geq 1$, by Minkowski’s inequality, for any $j=1,2,\dots,r$
we have

$\displaystyle\|G^{j}\|_{L_{q}(\Omega)}$
$\displaystyle=\left(\int_{\Omega}\left|\int_{I}f(x,t)W^{j}(t)\mathrm{d}t\right|^{q}\mathrm{d}x\right)^{1/q}$
$\displaystyle\leq\int_{I}\left(\int_{\Omega}\left|f(x,t)W^{j}(t)\right|^{q}\mathrm{d}x\right)^{1/q}\mathrm{d}t$
$\displaystyle=\int_{I}\left|W^{j}(t)\right|\left\|f(\cdot,t)\right\|_{L_{q}(\Omega)}\mathrm{d}t$
$\displaystyle\leq\left\|W^{j}(t)\right\|_{L_{2}(I)}\left\|\left\|f(\cdot,t)\right\|_{L_{q}(\Omega)}\right\|_{L_{2}(I)}$

so that

$\|G^{j}\|_{L_{q}(\Omega)}\lesssim\|f\|_{L_{2}(I,L_{q}(\Omega))},\quad
j=1,2,\dots,r.$ (57)

We now deal with $\left|G^{j}\right|_{B^{s}_{q,q}(\Omega)}$. Observe that for
any $j$ we have

$\displaystyle\left|G^{j}\right|_{B^{s}_{q,q}(\Omega)}$
$\displaystyle\lesssim\left(\int_{0}^{1}\left[u^{-s}w_{r}(G^{j},I,u)_{q}\right]^{q}\frac{\mathrm{d}u}{u}\right)^{1/q}=\left(\int_{0}^{1}u^{-sq}w_{r}(G^{j},I,u)_{q}^{q}\frac{\mathrm{d}u}{u}\right)^{1/q}$
$\displaystyle=\left(\int_{0}^{1}u^{-sq}\frac{1}{(2u)^{n}}\int_{|h|\leq
u}\left\|\Delta_{h}^{r}G^{j}\right\|^{q}_{L_{q}(\Omega_{rh})}\mathrm{d}h\frac{\mathrm{d}u}{u}\right)^{1/q}$
$\displaystyle=\left(\int_{0}^{1}u^{-sq}\frac{1}{(2u)^{n}}\int_{|h|\leq
u}\int_{\Omega_{rh}}\left|\int_{I}\Delta_{h}^{r}f(t,x)W^{j}(t)\mathrm{d}t\right|^{q}\mathrm{d}x\mathrm{d}h\frac{\mathrm{d}u}{u}\right)^{1/q}$
$\displaystyle=\left(\int_{0}^{1}\int_{|h|\leq
u}\int_{\Omega_{rh}}u^{-sq}\frac{1}{(2u)^{n}}\left|\int_{I}\Delta_{h}^{r}f(t,x)W^{j}(t)\mathrm{d}t\right|^{q}\mathrm{d}x\mathrm{d}h\frac{\mathrm{d}u}{u}\right)^{1/q}.$

Again, by Minkowski’s inequality

$\displaystyle\left|G^{j}\right|_{B^{s}_{q,q}(\Omega)}$
$\displaystyle\leq\int_{I}\left(\int_{0}^{1}\int_{|h|\leq
u}\int_{\Omega_{rh}}u^{-sq}\frac{1}{(2u)^{n}}\left|\Delta_{h}^{r}f(t,x)W^{j}(t)\right|^{q}\mathrm{d}x\mathrm{d}h\frac{\mathrm{d}u}{u}\right)^{1/q}\mathrm{d}t$
$\displaystyle=\int_{I}\left|W^{j}(t)\right|\left(\int_{0}^{1}u^{-sq}\frac{1}{(2u)^{n}}\int_{|h|\leq
u}\int_{\Omega_{rh}}\left|\Delta_{h}^{r}f(t,x)\right|^{q}\mathrm{d}x\mathrm{d}h\frac{\mathrm{d}u}{u}\right)^{1/q}\mathrm{d}t$
$\displaystyle=\int_{I}\left|W^{j}(t)\right|\left|f(t)\right|_{B^{s}_{q,q}(\Omega)}\mathrm{d}t\leq\left\|W^{j}\right\|_{L_{2}(I)}\left\|f(t)\right\|_{L_{2}(I,B^{s}_{q,q}(\Omega))}$

whence

$\left|G^{j}\right|_{B^{s}_{q,q}(\Omega)}\lesssim\left\|f\right\|_{L_{2}(I,B^{s}_{q,q}(\Omega))}.$
(58)

Therefore from (57) (58) and (56) we get

$\|G\|_{L_{2}(I,B^{s}_{q,q}(\Omega))}\lesssim\|f\|_{L_{2}(I,B^{s}_{q,q}(\Omega))}\qquad\text{if
\ $1\leq q<\infty$},$

and analogously for $q=\infty$. ∎

If we use the same polynomial degree in space and time in Theorem 25 the
result reads as follows.

######  Corollary 28 (Fully discrete with same polynomial degree).

Let $1\leq q\leq\infty$ and
$n\Big{(}\frac{1}{q}-\frac{1}{2}\Big{)}_{+}<s<r\in\mathbb{N}$. If $f\in
B^{s}_{q,q}([0,T),X)\cap L_{2}([0,T),B_{q,q}^{s}(\Omega))$ with
$X=L_{2}(\Omega)$, then for each $\varepsilon>0$ there exists a time-space
partition $\mathcal{P}$ that satisfies

$\\#\mathcal{P}\leq c_{1}\varepsilon^{-\frac{n+1}{s}}$

and a function $F\in\overline{\mathbb{V}}_{\mathcal{P}}^{r,r}$ such that

$\displaystyle\|f-F\|_{L_{2}([0,T),X)}\leq
c_{2}\varepsilon|\\!|\\!|f|\\!|\\!|\leq
c_{3}(\\#\mathcal{P})^{-\frac{s}{n+1}}|\\!|\\!|f|\\!|\\!|,$

where
$|\\!|\\!|f|\\!|\\!|=|f|_{B^{s}_{q,q}([0,T),X)}+\|f\|_{L_{2}([0,T),B_{q,q}^{s}(\Omega))}$
and the positive constants $c_{1},c_{2},c_{3}$ depend on $q$ and $s$ but not
on $f$.

###  5.2. Comparison with space-time finite elements

If we were to use space-time finite elements of order $r$ in n+1, in order to
obtain the same rate $(\\#\mathcal{P})^{-\frac{s}{n+1}}$ as that indicated in
Corollary 28, Corollary 23 tells us that the function $f$ should belong to
$B^{s}_{q,q}([0,T)\times\Omega)$ with $0<s<r$ and
$0<\frac{1}{q}<\frac{s}{n+1}+\frac{1}{2}$. This raises the following question:

> What is the relation between the spaces
>
> $B^{s}_{q,q}([0,T)\times\Omega)\quad\text{and}\quad
> B^{s}_{q_{1},q_{1}}([0,T),L_{2}(\Omega))\cap
> L_{2}([0,T),B_{q_{2},q_{2}}^{s}(\Omega))$
>
> for the respective ranges of the parameters $q_{1},q_{2},$ and $q$?

The following proposition provides a first attempt to give an answer to this
question.

######  Proposition 29.

Let $0<s<r$ and $0<q_{1},q_{2},q<\infty$, where we additionally require that

$\frac{1}{q}<\frac{s}{n+1}+\frac{1}{2},\qquad\frac{1}{q_{1}}<{s}+\frac{1}{2},\qquad\text{and}\qquad\frac{1}{q_{2}}<\frac{s}{n}+\frac{1}{2}.$
(59)

Then we have

$\bigcup_{q_{1},q_{2}}B^{s}_{q_{1},q_{1}}([0,T),L_{2}(\Omega))\cap
L_{2}([0,T),B_{q_{2},q_{2}}^{s}(\Omega))\not\subset\bigcup_{q}B^{s}_{q,q}([0,T)\times\Omega),$
(60)

where the union is taken over all $q,q_{1},q_{2}$ according to (59).

###### Proof.

We show that we can find functions belonging to
$\bigcup_{q_{1},q_{2}}B^{s}_{q_{1},q_{1}}([0,T),L_{2}(\Omega))\cap
L_{2}([0,T),B_{q_{2},q_{2}}^{s}(\Omega))$ which are not in
$\bigcup_{q}B^{s}_{q,q}([0,T)\times\Omega)$. For this let us choose $q_{1}$
such that

$\frac{1}{q}<\frac{s}{n+1}+\frac{1}{2}<\frac{1}{q_{1}}<s+\frac{1}{2}$

and consider a function $f$ which is constant with respect to the space
variable $x$ and belongs to $B^{s}_{q_{1},q_{1}}([0,T))$. Clearly, by our
assumptions this function is also in $L_{2}([0,T))$. Moreover, by or choice of
$q_{1}$ we see from [HS13, Cor. 3.7] that

$B^{s}_{q_{1},q_{1}}([0,T),L_{2}(\Omega))\not\hookrightarrow
B^{s}_{q,q}([0,T)\times\Omega),$

since $q_{1}<q$, which proves the claim. Alternatively, we could choose
$q_{2}$ such that

$\frac{1}{q}<\frac{s}{n+1}+\frac{1}{2}<\frac{1}{q_{2}}<\frac{s}{n}+\frac{1}{2}$

and consider a function $f$ which is constant with respect to the time
variable $t$ and belongs to $B^{s}_{q_{2},q_{2}}(\Omega)$. Then, clearly this
function also belongs to $L_{2}(\Omega)$. By our choice of $q_{2}$ it again
follows from [HS13, Cor. 3.7] that

$B^{s}_{q_{2},q_{2}}(\Omega)\not\hookrightarrow B^{s}_{q,q}(\Omega),$

since $q_{2}<q$. This completes the proof. ∎

######  Remark 30.

We believe that for the above spaces under consideration we actually have the
following inclusion

$\bigcup_{q}B^{s}_{q,q}([0,T)\times\Omega)\subsetneq\bigcup_{q_{1},q_{2}}B^{s}_{q_{1},q_{1}}([0,T),L_{2}(\Omega))\cap
L_{2}([0,T),B_{q_{2},q_{2}}^{s}(\Omega)),$

where $q,q_{1}$, and $q_{2}$ are chosen according to (59). This can be
interpreted in the sense that the respective solution spaces for time-stepping
algorithms yielding the approximation class
$\mathbb{A}_{\frac{s}{n+1}}(L_{2}([0,T)\times\Omega))$ are in fact larger than
the corresponding solution spaces for space-time finite elements.
In this context the fact that $B^{s}_{q,q}([0,T)\times\Omega)\subset
L_{2}([0,T),B^{s}_{q_{2},q_{2}}(\Omega))$ should be easier to handle. However,
these matters are quite technical and, therefore, this interesting question
will be tackled in a future paper.

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Marcelo Actis, Universidad Nacional del Litoral and CONICET, Departamento de
Matemática, Faculdad de Ingeniería Química, S3000AOM Santa Fe, Argentina
Phone: (+54) 342 457 1164
E-mail<EMAIL_ADDRESS>

Pedro Morin, Universidad Nacional del Litoral and CONICET, Departamento de
Matemática, Faculdad de Ingeniería Química, S3000AOM Santa Fe, Argentina
Phone: (+54) 342 457 1164
E-mail<EMAIL_ADDRESS>

Cornelia Schneider, Friedrich-Alexander-Universität Erlangen-Nürnberg,
Lehrstuhl AM3, Cauerstr. 11, 91058 Erlangen, Germany
Phone: (+49) 9131 85-67207, Fax: (+49) 9131 85-67201
E-mail<EMAIL_ADDRESS>