# To Multidimensional Mellin Analysis: Besov spaces, $K$-functor,
approximations, frames

Isaac Z. Pesenson  Department of Mathematics, Temple University, Philadelphia,
PA 19122<EMAIL_ADDRESS>Dedicated to 95th birthday of my teacher Paul
L.Butzer

###### Abstract.

In the setting of the multidimensional Mellin analysis we introduce moduli of
continuity and use them to define Besov-Mellin spaces. We prove that Besov-
Mellin spaces are the interpolation spaces (in the sense of J.Peetre) between
two Sobolev-Mellin spaces. We also introduce Bernstein-Mellin spaces and prove
corresponding direct and inverse approximation theorems. In the Hilbert case
we discuss Laplace-Mellin operaor and define relevant Paley-Wiener-Mellin
spaces. Also in the Hilbert case we describe Besov-Mellin spaces in terms of
Hilbert frames.

###### Key words and phrases:

Multidimensioal Mellin analysis, Sobolev-Mellin, Besov-Mellin, Bernsten-Mellin
spaces, K-functor, one-parameter groups of operators, modulus of continuity,
Laplace-Mellin operator, Paley-Wiener functions, direct and inverse
approximation theorems, frames.

###### 1991 Mathematics Subject Classification:

43A85, 41A17;

##  1\. Introduction

By one-dimensional Mellin analysis we understand the following setup. For
$p\in[1,\infty[$, denote by $\|\cdot\|_{p}$ the norm of the Lebesgue space
$L^{p}(\mathbb{R}_{+}).$ In Mellin analysis, the analogue of
$L^{p}(\mathbb{R}_{+})$ are the spaces $X^{p}(\mathbb{R}_{+})$ comprising all
functions $f:\mathbb{R}_{+}\mapsto\mathbb{C}$ such that
$f(\cdot)(\cdot)^{-1/p}\in L^{p}(\mathbb{R}_{+})$ with the norm
$\|f\|_{X^{p}(\mathbb{R}_{+})}:=\|f(\cdot)(\cdot)^{-1/p}\|_{p}$. Furthermore,
for $p=\infty$, we define $X^{\infty}$ as the space of all measurable
functions $f:\mathbb{R}_{+}\mapsto\mathbb{C}$ such that
$\|f\|_{X^{\infty}}:=\sup_{x>0}|f(x)|<\infty$.

In spaces $X^{p}(\mathbb{R}_{+})$ we consider the one-parameter $C_{0}$-group
of operators $U(t),\>t\in\mathbb{R},$ where

(1.1) $U(t)f(x)=f(e^{t}x),\>\>\>U(t+\tau)=U(t)U(\tau),$

whose infinitesimal generator is

(1.2) $\frac{d}{dt}U(t)f(x)|_{t=0}=x\frac{d}{dx}f(x)=\Theta f(x).$

The described setting closely related to the Mellin transform

(1.3) $\mathcal{M}f(s)=\int_{0}^{\infty}f(x)x^{s-1}dx,$

and its inverse

(1.4) $f(x)=\frac{1}{2\pi
i}\int_{c-i\infty}^{c+i\infty}\mathcal{M}f(s)x^{-s}ds,\>\>\>c\in\mathbb{R}.$

This pair of transforms has many deep and fascinating connection to many
fundamental notions in analysis. For a detailed account of the Mellin Harmonic
Analysis, its history, and connections to other mathematical ideas see the
beautiful paper [15]. A lot of relevant information can also be found in
interesting articles [2], [17]. In particular, in a series of papers [3]-[11],
[16], [37] authors developed in the framework of one-dimensional Mellin
analysis analogs of such important topics as Bernstein spaces, Bernstein
inequality, Paley-Wiener theorem, Riesz-Boas interpolation formulas, different
sampling results including exponential sampling.

The present paper has two main objectives: 1) to go from one-dimensional
setting to multidimensional, and 2) to concentrate on such important topics as
Besov spaces, $K$-functor, approximations, frames. As well as we know, such
topics were never discussed in the setting of the multidimensional Mellin
analysis.

The paper is organized as follows. In section 2 we desribe the basic
assumptions and notations of multidimensional Mellin analysis, In section 3 in
the setting of the multidimensional Mellin analysis we introduce moduli of
continuity and use them to define Besov-Mellin spaces. In section 4 we prove
that Besov-Mellin spaces are the interpolation spaces (in the sense of
J.Peetre) between two Sobolev-Mellin spaces. In section 5 we introduce
Bernstein-Mellin spaces and in section 6 proove corresponding direct and
inverse approximation theorems including a Jackson-type inequality. Sections
7-9 devoted to the Hilbert case. We introduce and discuss Laplace-Mellin
operaor in section 7. In section 8 we define relevant Paley-Wiener spaces and
in section 9 consider Hilbert frames. In section 10 we describe Besov-Mellin
spaces in terms of approximations by Paley-Wiener functions and in terms of
some Hilbert frames. Throughout the paper we extensively use the Peetre’s
theory of interpolation and approximation spaces. To make the paper self
contained some related definitions, facts and proofs are collected in
Appendices $1$ and $2.$

All our statements concerning the multidimensional Mellin analysis seems to be
new. However, at least some of them are just adaptations and particular
realizations of previously obtained (but maybe not very well known) more
general results (see [14], [26]-[36], [39]). In this sense the article can be
treated as a review paper. Our main goal was to attract attention to
multidimensional Mellin analysis, which is currently underdeveloped.

##  2\. Multidimensional Mellin Analysis

###  2.1. Space $X^{p}=X^{p}(\mathbb{R}_{+}^{n},d\mu)$. Mellin translations
and their generators

We set $\mathbb{R}_{+}=\\{x\in\mathbb{R},\>x>0\\}$ and
$\mathbb{R}_{+}^{n}=\\{(x_{1},..,x_{n}),x_{j}\in\mathbb{R}_{+},\>j=1,..,n\\}$,
and introduce the measure $d\mu=\frac{dx_{1},...dx_{n}}{x_{1}...x_{n}}$. The
corresponding space $L^{p},\>1\leq p<\infty$ of complex valued functions
defined on $\mathbb{R}_{+}^{n}$ is denoted as
$X^{p}=X^{p}(\mathbb{R}_{+}^{n},d\mu)$ and its norm is

$\|f\|_{X^{p}}=\left(\int_{\mathbb{R}_{+}^{n}}|f|^{p}d\mu\right)^{1/p},$

for $1\leq p<\infty$ We define $X^{\infty}$ as the space of all measurable
functions $f:\mathbb{R}_{+}^{n}\mapsto\mathbb{C}$ such that

$\|f\|_{X^{\infty}}:=ess\>sup_{x=(x_{1},...,x_{n})\in\mathbb{R}_{+}^{n}}|f(x)|<\infty.$

In spaces $X^{p},\>\>1\leq p<\infty,$ we consider the one-dimensional
$C_{0}$-groups of operators $U_{j}(t),\>t\in\mathbb{R},\>j=1,...,n,$ where

(2.1) $U_{j}(t)f(x_{1},..,x_{n})=f(x_{1},...,,e^{t}x_{j},...,x_{n}),$

whose infinitesimal generator is
$\>\Theta_{j}=x_{j}\partial_{j},\>\>j=1,...,n,$ since

(2.2) $\lim_{t\rightarrow
0}\frac{1}{t}\left(U_{j}(t)f(x_{1},...,x_{n})-f(x_{1},...,x_{n})\right)=\frac{d}{dt}U(t)f(x)|_{t=0}=x_{j}\partial_{j}f(x).$

The general theory (see [14], [20]) implies that a such operator is closed in
$X^{p},\>\>1\leq p<\infty,$. Its domain $\mathcal{D}(\Theta_{j})$ is dense in
$X^{p},\>\>1\leq p<\infty,$ and it consists of all functions in $X^{p}$ for
which the limit (2.2) exists in the norm of $X^{p},\>\>1\leq p<\infty$.
Domains of the powers of $\Theta_{j}$ are
$\mathcal{D}(\Theta_{j}^{k}),\>\>k\in\mathbb{N},$ and
$\mathcal{D}^{\infty}(\Theta_{j})=\cap_{k=1}^{\infty}\mathcal{D}(\Theta_{j}^{k})$.
We will also use the notation

$\mathcal{D}^{\infty}=\mathcal{D}^{\infty}(\Theta_{1},...,\Theta_{n})=\cap_{k\in\mathbb{N}}\cap_{|\alpha|=k}\mathcal{(}\Theta_{1}^{\alpha_{1}}...\Theta_{n}^{\alpha_{n}}),\>\>\>\alpha=(\alpha_{1},...,\alpha_{n}),\>\>\alpha_{j}\in\mathbb{N},\>\>1\leq
j\leq n.$

We note that the groups $U_{j}$ commute with each other

$U_{i}(t_{i})U_{j}(t_{j})f=U_{j}(t_{j})U_{i}(t_{i})f,\>\>\>\>f\in
X^{p},\>\>1\leq p<\infty,\>\>1\leq i,j\leq n,\>\>t_{i},t_{j}\geq 0.$

Also, one can easily verify that $\mathcal{D}^{\infty}$ is invariant under all
$U_{j}$ and all $\Theta_{j}$ and

(2.3)
$\Theta_{j}U_{1}(t_{1})U_{2}(t_{2})...U_{n}(t_{n})f=U_{1}(t_{1})U_{2}(t_{2})...\Theta_{j}U_{j}(t_{j})...U_{n}(t_{n})f,\>\>\>f\in\mathcal{D}^{\infty}.$

One can also check that for the

$Uf(t_{1},t_{2},...,t_{n})=U_{1}(t_{1})U_{2}(t_{2})...U_{n}(t_{n}),\>\>\>(t_{1},...,t_{n})\in\mathbb{R}^{n},$

the following formula holds

(2.4)
$\Theta_{j}Uf(t_{1},...,t_{n})=\partial_{j}\left[Uf(t_{1},...,t_{n})\right],\>\>\>f\in\mathcal{D}^{\infty}.$

##  3\. Sobolev-Mellin and Besov-Mellin spaces

######  Definition 1.

The Banach space
$\mathcal{W}^{k}_{p}=\mathcal{W}^{k}_{p}(\mathbb{R}^{n}_{+},d\mu),\>\>k\in\mathbb{N},,\>\>1\leq
p<\infty,$ is the set of functions $f$ in
$X^{p}=X^{p}(\mathbb{R}_{+}^{n},d\mu)$ for which the following norm is finite

$\|f\|_{\mathcal{W}^{k}_{p}}=\|f\|_{X^{p}}+\sum_{|\alpha|=k}\|\Theta_{1}^{\alpha_{1}}...\Theta_{n}^{\alpha_{n}}\|_{X^{p}},$

where
$\alpha=(\alpha_{1},...,\alpha_{n}),\>|\alpha|=\alpha_{1}+...+\alpha_{n},\>\alpha_{1},...,\alpha_{n}\in\mathbb{N}$.

By using closeness of the operators $\Theta_{j}$ one can check that the norm
$\|f\|_{\mathcal{W}^{k}_{p}}$ is equivalent to the norm

$|||f|||_{\mathcal{W}^{k}_{p}}=\|f\|_{X^{p}}+\sum_{k=1}^{r}\sum_{1\leq
j_{1},...j_{k}\leq n}\|\Theta_{j_{1}}...\Theta_{j_{k}}f\|_{X^{p}},$

where $r\in\mathbb{N},\>\>f\in X^{p},\>\>1<p<\infty.$

One has the following version of the Kolmogorov-Stein-Mellin inequality which
follows from a general fact Theorem 12.3 in Appendix 2.

######  Theorem 3.1.

The following holds for $f\in\mathcal{D}(\Theta_{j}^{m}),\>\>1\leq p<\infty,$

(3.1) $\left\|\Theta_{j}^{k}f\right\|_{X^{p}}^{m}\leq
C_{k,m}\|\Theta_{j}^{m}f\|_{X^{p}}^{k}\|f\|_{X^{p}}^{m-k},\>\>\>0\leq k\leq
m,$

where $C_{k,m}=(K_{m-k})^{m}/(K_{m})^{m-k},$ and

$K_{j}=\frac{4}{\pi}\sum_{r=0}^{\infty}\frac{(-1)^{r(j+1)}}{(2r+1)^{j+1}},\>\>\>j,\>\>r\in\mathbb{N}.$

is a Favard constant.

The mixed modulus of continuity is introduced as

$\Omega^{r}_{p}(s,f)=$ (3.2) $\sum_{1\leq j_{1},...,j_{r}\leq
n}\sup_{0\leq\tau_{j_{1}}\leq s}...\sup_{0\leq\tau_{j_{r}}\leq
s}\|\left(U_{j_{1}}(\tau_{j_{1}})-I\right)...\left(U_{j_{r}}(\tau_{j_{r}})-I\right)f\|_{X^{p}},$

where $f\in X^{p},\ r\in\mathbb{N},$ and $I$ is the identity operator in
$X^{p},\>\>1\leq p<\infty.$

######  Definition 2.

For $\alpha\in\mathbb{R}_{+},\>\>1\leq p<\infty,\>\>1\leq q\leq\infty,$ and
$\alpha$ is not integer the Besov-Mellin space $\mathcal{B}^{\alpha}_{p,q}$ is
defined as the subspace in $X^{p}$ of all the functions for which the
following norm is finite

(3.3) $\|f\|_{\mathcal{W}^{[\alpha]}_{p}}+\sum_{1\leq
j_{1},...,j_{[\alpha]}\leq
n}\left(\int_{0}^{\infty}\left(s^{[\alpha]-\alpha}\Omega^{1}_{p}(s,\Theta_{j_{1}}...\Theta_{j_{[\alpha]}}f)\right)^{q}\frac{ds}{s}\right)^{1/q},$

where $[\alpha]$ is the integer part of $\alpha$.

In the case when $\alpha=k\in\mathbb{N}$ is an integer the Besov-Mellin space
$\mathcal{B}^{\alpha}_{p,q},\>\>1\leq p<\infty,\>\>1\leq q\leq\infty,$ is
defined as the subspace in $X^{p}$ of all the functions for which the
following norm is finite (Zygmund condition)

(3.4) $\|f\|_{\mathcal{W}^{k-1}_{p}}+\sum_{1\leq j_{1},...,j_{k-1}\leq
n}\left(\int_{0}^{\infty}\left(s^{-1}\Omega^{2}_{p}(s,\Theta_{j_{1}}...\Theta_{j_{k-1}}f)\right)^{q}\frac{ds}{s}\right)^{1/q}.$

##  4\. The interpolation theorem for the Sobolev-Mellin and Besov-Mellin
spaces

It is well known that the most most important property of the classical Besov
spaces is the fact that they are interpolation spaces (see section 11 below)
between two Sobolev spaces [14], [21]. Here we formulate this result for
Besov-Mellin spaces.

For the pair of Banach spaces $(X^{p},\mathcal{W}_{p}^{r}),,\>\>1\leq
p<\infty,\>\>$ the $K$-functor is defined by the formula (see section 11)

$K(s^{r},f,X^{p},\mathcal{W}^{r}_{p})=\inf_{f=f_{0}+f_{1},f_{0}\in
X^{p},f_{1}\in\mathcal{W}_{p}^{r}}\left(\|f_{0}\|_{X^{p}}+s^{r}\|f_{1}\|_{\mathcal{W}_{p}^{r}}\right).$

The results below are the particular case of the results in [27], [36] (see
also [22], [26]-[28], [30]) which correspond to a situation with commuting
one-parameter semigroups.

We consider the so-called Hardy-Steklov-Mellin operator $\mathcal{P}_{r}(s)$
which is defined on $X^{p},\>\>1\leq p<\infty,$ by the formula

$\mathcal{P}_{r}(s)=\mathcal{P}_{1,r}(s)\mathcal{P}_{2,r}(s)...\mathcal{P}_{n,r}(s)f,\>\>\>f\in
X^{p},\>\>1\leq p<\infty,$

where

$\mathcal{P}_{j.r}(s)=(s/r)^{-r}\int_{0}^{s/r}...\int_{0}^{s/r}\sum_{k=1}^{r}(-1)^{k}C^{k}_{r}U_{j}(k(\tau_{j,1}+...+\tau_{j,r})fd\tau_{j,1}...d\tau_{j,r},$

and $C^{k}_{r}$ are the binomial coefficients. As it is shown in [27], [36],

$\mathcal{P}_{r}(s):X^{p}\mapsto\mathcal{W}_{p}^{r},\>\>1\leq
p<\infty,\>r\in\mathbb{N},\>s\in\mathbb{R}.$

Moreover, there exist constants $c>0,\>C>0$, such that for all $f\in
X^{p},\>\>s\geq 0,\>\>1\leq p<\infty,\>\>$

(4.1) $c\>\Omega^{r}_{p}(s,f)\leq K(s^{r},f,X^{p},\>\mathcal{W}^{r}_{p})\leq
C\left(\Omega^{r}_{p}(s,f)+\min(s^{r},1)\|f\|_{X^{p}}\right).$

This inequality implies density of $\mathcal{W}^{r}_{p},\>\>1\leq p<\infty,\>$
in $X^{p}$.

######  Theorem 4.1.

The following holds true.

  1. (1)

The Besov-Mellin space $\mathcal{B}^{\alpha}_{pq}$ coincides with the
interpolation space
$\left(X^{p},\>\mathcal{W}^{r}_{p}\right)^{K}_{\alpha/r,q},\
0<\alpha<r\in\mathbb{N},\ 1\leq p<\infty,\>1\leq q\leq\infty,$ and its norm
(3.3), (3.4) is equivalent to either of the following norms

(4.2)
$\|f\|_{X^{p}}+\left(\int_{0}^{\infty}(s^{-\alpha}\Omega^{r}_{p}(s,f))^{q}\frac{ds}{s}\right)^{1/q},\>\>\>1\leq
q<\infty,$

or to

$\|f\|_{X^{p}}+\left(\int_{0}^{\infty}\left(s^{-\alpha}K(s^{r},f,X^{p},\>\mathcal{W}^{r}_{p})\right)^{q}\frac{ds}{s}\right)^{1/q},\>\>\>1\leq
q<\infty,$

with the usual modifications for $q=\infty$.

  2. (2)

The following isomorphism holds true

$\left(X^{p},\>\mathcal{W}^{r}_{p}\right)^{K}_{\alpha/r,q}=\left(\mathcal{W}^{k_{1}}_{p},\>\mathcal{W}^{k_{2}}_{p}\right)^{K}_{(\alpha-
k_{1})/(k_{2}-k_{1}),q},$

where $0\leq k_{1}<\alpha<k_{2}\leq r\in\mathbb{N},\>,\>\>1\leq
p<\infty,\>\>1\leq q\leq\infty.$

##  5\.  Bernstein-Mellin spaces

Let’s remind that in the classical analysis a Bernstein class [1], [24], which
is denoted as ${\bf B}_{\sigma}^{p}(\mathbb{R}),\>\>\sigma\geq 0,\>\>1\leq
p\leq\infty,$ is a linear space of all functions
$f:\mathbb{R}\mapsto\mathbb{C}$ which belong to $L^{p}(\mathbb{R})$ and admit
extension to $\mathbb{C}$ as entire functions of exponential type $\sigma$. It
is known, that for every such function there exists a constant $A>0$ such the
following inequality holds

$|f(x+iy)|\leq Ae^{\sigma|y|}.$

A function $f$ belongs to ${\bf B}_{\sigma}^{p}(\mathbb{R})$ if and only if
the following Bernstein inequality holds

$\|f^{(k)}\|_{L^{p}(\mathbb{R})}\leq\sigma^{k}\|f\|_{L^{p}(\mathbb{R})}$

for all natural $k$. Using the distributional Fourier transform

$\widehat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}f(x)e^{-i\xi
x}dx,\>\>\>f\in L^{p}(\mathbb{R}),\>\>1\leq p\leq\infty,$

one can show (Paley-Wiener theorem) that $f\in{\bf
B}_{\sigma}^{p}(\mathbb{R}),\>\>1\leq p\leq\infty,$ if and only if $f\in
L^{p}(\mathbb{R}),\>\>1\leq p\leq\infty,$ and the support of $\widehat{f}$ (in
sens of distributions) is in $[-\sigma,\sigma]$.

######  Definition 3.

For any natural $1\leq k\leq n$ the Bernstein-Mellin space ${\bf
B}_{\sigma}^{p}(\Theta_{1},...,\Theta_{n}),\>\>\sigma>0,\>1\leq p<\infty,$ is
defined as a set of all functions $f$ in $X^{p}$ which belong to
$\mathcal{D}^{\infty}(\Theta)$ and for which

(5.1)
$\|\Theta_{j_{1}}^{l_{1}}...\Theta_{j_{k}}^{l_{k}}f\|_{X^{p}}\leq\sigma^{|l|}\|f\|_{X^{p}},$

where
$\>l=(l_{j_{1}},...,l_{j_{k}}),\>|l|=l_{j_{1}}+...+l_{j_{k}},\>l_{j}\in\mathbb{N}\cup\\{0\\}.$

The notation ${\bf B}_{\sigma}^{p}(\Theta_{j}),\>1\leq p<\infty,$ will be used
for the set of all functions $f$ in $X^{p}$ which belong to
$\mathcal{D}^{\infty}(\Theta_{j})$ and for which

(5.2) $\|\Theta_{j}^{l}f\|_{X^{p}}\leq\sigma^{l}\|f\|_{X^{p}}.$

The next Lemma follows from a more general result Lemma 12.1 in Appendix 2.

######  Lemma 5.1.

A function $f\in\mathcal{D}^{\infty}$ belongs to ${\bf
B}_{\sigma}^{p}(\Theta_{j}),\>1\leq j\leq n,>\>\sigma>0,\>1\leq p<\infty,$ if
and only if the quantity

(5.3) $\sup_{k\in
N}\sigma^{-k}\|\Theta_{j}^{k}f\|_{X^{p}(\mathbb{R}_{+})}=R(f,\sigma)$

is finite.

######  Theorem 5.2.

A function $f\in\mathcal{D}^{\infty}$ belongs to ${\bf
B}_{\sigma}^{p}(\Theta_{1},...,\Theta_{n}),\>\>\sigma>0,\>1\leq p<\infty,$ if
and only if the quantity

(5.4) $\sup_{|l|\in
N}\>\>\sup_{l=(l_{j_{1}},...,l_{j_{k}})}\sigma^{-|l|}\|\Theta_{j_{1}}^{l_{j_{1}}}...\Theta_{j_{k}}^{l_{j_{k}}}f\|_{X^{p}}=R(f,\sigma),\>\>\>|l|=l_{j_{1}}+...+l_{j_{k}},$

is finite.

###### Proof.

It is clear that if $f\in{\bf B}_{\sigma}^{p}(\Theta_{1},...,\Theta_{n})$ then
(5.4) holds. Now, we assume that (5.4) holds and we have to verify the
inequality (5.2) with $k=n$. The inequality (5.4) and the Lemma 5.1 imply that
for every $1\leq j\leq n$ the function $f$ belongs to ${\bf
B}_{\sigma}^{p}(\Theta_{j})$, i.e.

$\|\Theta_{j}^{\alpha}f\|_{X^{p}}\leq\sigma^{\alpha}\|f\|_{X^{p}},\>\alpha\in\mathbb{N}.$

We proceed by induction. Assuming that we already proved that for some $1\leq
k<n$ we proved that (5.4) implies (5.2). Clearly, (5.4) implies that any
$\Theta_{j_{1}}^{\alpha_{1}}...\Theta_{j_{k}}^{\alpha_{k}}f$ satisfies (5.2)
for any $\Theta_{j}$ and Lemma 5.1 implies the inequality

$\|\Theta^{\alpha_{j_{k+1}}}_{j_{k+1}}\Theta_{j_{1}}^{\alpha_{1}}...\Theta_{j_{k}}^{\alpha_{k}}f\|_{X^{p}}\leq\sigma^{|\alpha_{j_{k+1}}|}\|\Theta_{j_{1}}^{\alpha_{1}}...\Theta_{j_{k}}^{\alpha_{k}}f\|_{X^{p}}\leq\sigma^{\left(|\alpha_{j_{k+1}}|+...+|\alpha_{j_{k}}|\right)}\|f\|_{X^{p}}.$

Theorem is proven.

∎

For a function $f\in\cup_{\sigma>0}\mathbf{B}_{\sigma}^{p}(\Theta_{j}),$ the
$\sigma_{j;f}$ will denote the smallest $\sigma$ for which the inequality
(5.2) holds. The next statement follows from the corresponding Theorem 12.4 in
our Appendix 2.

######  Theorem 5.3.

Let $f\in X^{p},\>1\leq p<\infty,$ belongs to a space
$\mathbf{B}_{\sigma}^{p}(\Theta_{j}),$ for some $0<\sigma<\infty.$ Then the
following limit exists

(5.5) $d_{j;f}=\lim_{k\rightarrow\infty}\|\Theta_{j}^{k}f\|^{1/k}_{X^{p}},$

and $d_{j,f}=\sigma_{j,f}.$ Conversely, if
$f\in\mathcal{D}^{\infty}(\Theta_{j})$ and
$d_{j,f}=\lim_{k\rightarrow\infty}\|\Theta_{j}^{k}f\|^{1/k}_{X^{p}},$ exists
and is finite, then $f\in\mathbf{B}^{p}_{d_{j,f}}(\Theta_{j})$ and
$d_{j,f}=\sigma_{j,f}.$

##  6\. Direct and inverse approximation theorems in Mellin analysis

The lemma below is a particular case of the Lemma 12.2 in Appendix 2.

######  Lemma 6.1.

[35] Suppose that $p\in{\bf B}^{1}_{\sigma}(\mathbb{R})$ is an entire function
of exponential type $\sigma$. Then for any $f\in X^{p}$ the function

$P_{j,\sigma}f=\int_{-\infty}^{\infty}p(t)U_{j}(t)fdt$

belongs to ${\bf B}_{\sigma}^{p}(\Theta_{j}).$ The integral here is understood
in the sense of Bochner for the abstract-valued function
$p(t)U_{j}(t)f:\mathbb{R}\mapsto X^{p},\>1\leq p<\infty$.

######  Theorem 6.2.

Let $p\in L_{1}(\mathbb{R}),\>\>\|f\|_{L_{1}(\mathbb{R})}=1,$ is an entire
function of exponential type $\sigma$ then for any $f\in X^{p},\>1\leq
p<\infty,$ the function

$P_{\sigma}f=\underbrace{\int_{-\infty}^{\infty}...\int_{-\infty}^{\infty}}_{n}p(\tau_{1})...p(\tau_{n})U_{1}(\tau_{1})...U_{n}(\tau_{n})fd\tau_{1}..d\tau_{n}$

belongs to ${\bf B}_{\sigma}^{p}(\Theta_{1},...,\Theta_{n}),\>1\leq p<\infty$.
The integral here is understood in the sense of Bochner for the abstract-
valued function
$p(\tau_{1})...p(\tau_{n})U_{1}(\tau_{1})...U_{n}(\tau_{n})f:\mathbb{R}^{n}\mapsto
X^{p}$.

###### Proof.

Indeed, due to commutativity and the previous Lemma one has

$\|\Theta_{j_{1}}^{l_{1}}\Theta_{j_{2}}^{l_{2}}...\Theta_{j_{k}}^{l_{k}}P_{\sigma}f\|_{X^{p}}=\left\|\Theta_{j_{1}}^{l_{1}}\int_{-\infty}^{\infty}p_{j_{1}}U_{j_{1}}(\tau_{j_{1}})\left[T_{j_{1}}(f)\right]d\tau_{j_{1}}\right\|_{X^{p}}\leq\sigma^{l_{1}}\|T_{j_{1}}(f)\|_{X^{p}},$

where

$T_{j_{1}}(f)=$
$\Theta_{j_{2}}^{l_{2}}...\Theta_{j_{k}}^{l_{k}}\underbrace{\int_{-\infty}^{\infty}...\int_{-\infty}^{\infty}}_{n-1}p(\tau_{1})...\widehat{p(\tau_{j_{1}})}...p(\tau_{n})U_{1}(\tau_{1})...\widehat{U_{j_{1}}(\tau_{j_{1}})}...U_{n}(\tau_{n})fd\tau_{1}...\widehat{d\tau_{j_{1}}}...d\tau_{n},$

where a wedge means that the corresponding term is missing. It is obvious that
continue this way we obtain

$\|\Theta_{j_{1}}^{l_{1}}\Theta_{j_{2}}^{l_{2}}...\Theta_{j_{k}}^{l_{k}}P_{\sigma}f\|_{X^{p}}\leq\sigma^{|l|}\|f\|_{X^{p}}.$

Applying Theorem 5.2 we finish the proof. ∎

Let’s introduce the functionals

$\omega_{j,p}^{m}\left(s,f\right)=\sup_{0\leq\tau\leq
s}\|\left(U_{j}(\tau)-I\right)^{m}f\|_{X^{p}},$
$\mathcal{E}_{j,p}(\sigma,f)=\inf_{g\in{\bf
B}_{\sigma}^{p}(\Theta_{j})}\|f-g\|_{X^{p}},$

where $1\leq p<\infty$. One has the following inequalities:

(6.1) $\omega_{j,p}^{m}\left(s,f\right)\leq
s^{k}\omega^{m-k}_{j,p}(s,\Theta_{j}^{k}f),\>\>\>\>0\leq k\leq m,$

and

(6.2)
$\omega_{j,p}^{m}\left(as,f\right)\leq\left(1+a\right)^{m}\omega_{j,p}^{m}(s,f),\>\>\>a\in\mathbb{R}_{+}.$

The first one follows from the identity

(6.3)
$\left(U_{j}(s)-I\right)^{k}f=\int_{0}^{s}...\int_{0}^{s}U_{j}(\tau_{1}+...+\tau_{k})\Theta_{j}^{k}fd\tau_{1}...d\tau_{k},$

where $I$ is the identity operator and $k\in\mathbb{N}$. The second one
follows from the property

$\omega_{j,p}^{1}\left(s_{1}+s_{2},f\right)\leq\omega_{j,p}^{1}\left(s_{1},f\right)+\omega_{j,p}^{1}\left(s_{2},f\right)$

which is easy to verify.

Our next objective is to prove an analog of the Jackson theorem. The proof of
the lemma below is motivated by Theorem 5.2.1 in [24].

Let

(6.4) $\rho(t)=a\left(\frac{\sin(t/N)}{t}\right)^{N}$

where $N=2(m+3)$ and

$a=\left(\int_{-\infty}^{\infty}\left(\frac{\sin(t/N)}{t}\right)^{N}dt\right)^{-1}.$

With such choice of $a$ and $N$ function $\rho$ will have the following
properties:

(1) $\rho$ is an even nonnegative entire function of exponential type one;

(2) $\rho$ belongs to $L_{1}(\mathbb{R})$ and its $L_{1}(\mathbb{R})$-norm is
$1$;

(3) the integral

(6.5) $\int_{-\infty}^{\infty}\rho(t)|t|^{m}dt$

is finite.

Next, we observe the following formula for every $1\leq j\leq n$

$(-1)^{m+1}(U_{j}(s)-I)^{m}f=$ (6.6)
$(-1)^{m+1}\sum^{m}_{k=0}(-1)^{m-k}C^{k}_{m}U_{j}(ks)f=\sum_{k=1}^{m}b_{k}U_{j}(ks)f-f,$

where $b_{1}+b_{2}+...+b_{m}=1.$ Consider the vector

(6.7)
$\mathcal{Q}_{j}(\sigma,m)(f)=\int_{-\infty}^{\infty}\rho(t)\left\\{(-1)^{m+1}\left(U_{j}\left(\frac{t}{\sigma}\right)-I\right)^{m}f+f\right\\}dt.$

According to (6) we have

$\mathcal{Q}_{j}(\sigma,m)(f)=\int_{-\infty}^{\infty}\rho(t)\sum_{k=1}^{m}b_{k}U_{j}\left(k\frac{t}{\sigma}\right)fdt.$

Changing variables in each of integrals

$\int_{-\infty}^{\infty}\rho(t)U_{j}\left(k\frac{t}{\sigma}\right)fdt,\>\>\>\>1\leq
k\leq m,$

we obtain the formula

$\mathcal{Q}_{j}(\sigma,m)(f)=\int_{-\infty}^{\infty}\Phi(t)U_{j}(t)fdt,$

where

$\Phi(t)=\sum_{k=1}^{m}b_{k}\left(\frac{\sigma}{k}\right)\rho\left(t\frac{\sigma}{k}\right),\>\>\>\>\>\>b_{1}+b_{2}+...+b_{m}=1.$

Since the function $\rho(t)$ has exponential type one every function
$\rho\left(t\frac{\sigma}{k}\right)$ has the type $\sigma/k,\>\>1\leq k\leq
m,$ and because of this the function $\Phi(t)$ has exponential type $\sigma$.
It also belongs to $L_{1}(\mathbb{R})$ and as it was just shown in the
previous statement it implies that the function $\mathcal{Q}_{j}(\sigma,m)(f)$
belongs to $\mathbf{B}_{p}^{\sigma}(\Theta_{j})$.

######  Lemma 6.3.

[35] For a given natural $m$ there exists a constant $c=c(m)>0$ such that for
all $\sigma>0$ and all $f$ in $X^{p},1\leq p<\infty,$

(6.8) $\mathcal{E}_{j,p}(\sigma,f)\leq
c\omega^{m}_{j,p}\left(1/\sigma,f\right).$

Moreover, for any $1\leq k\leq m$ there exists a $C=C(m,k)>0$ such that for
any $f\in\mathcal{D}(\Theta_{j}^{k})$ one has

(6.9)
$\mathcal{E}_{j,p}(\sigma,f)\leq\frac{C}{\sigma^{k}}\omega^{m-k}_{j,p}\left(1/\sigma,\>\Theta_{j}^{k}f\right),\>\>\>\>0\leq
k\leq m.$

###### Proof.

We estimate the error of approximation of $\mathcal{Q}_{j}(\sigma,m)(f)$ to
$f$. Since by (6.7)

$\mathcal{Q}_{j}(\sigma,m)(f)-f=(-1)^{m+1}\int_{-\infty}^{\infty}\rho(t)\left(U_{j}\left(\frac{t}{\sigma}\right)-I\right)^{m}fdt$

we obtain by using (6.2)

$\mathcal{E}_{j}(\sigma,f)\leq\left\|f-\mathcal{Q}_{j}(\sigma,m)(f)\right\|_{X^{p}}\leq\int_{-\infty}^{\infty}\rho(t)\left\|\left(U_{j}\left(\frac{t}{\sigma}\right)-I\right)^{m}f\right\|_{X^{p}}dt\leq$
(6.10)
$\int_{-\infty}^{\infty}\rho(t)\omega^{m}_{j,p}\left(t/\sigma,\>f\right)dt\leq
c\omega^{m}_{j,p}\left(1/\sigma,\>f\right),\>\>\>\>\>\>\>c=\int_{-\infty}^{\infty}\rho(t)(1+|t|)^{m}dt.$

If $f\in\mathcal{D}(\Theta_{j}^{k})$ then by using (6.1) we have

$\mathcal{E}_{j,p}(\sigma,f)\leq\int_{-\infty}^{\infty}\rho(t)\omega^{m}_{j,p}\left(t/\sigma,f\right)dt\leq$
(6.11)
$\frac{\omega^{m-k}_{j,p}\left(1/\sigma,\>\Theta_{j}^{k}f\right)}{\sigma^{k}}\int_{-\infty}^{\infty}\rho(t)|t|^{k}(1+|t|)^{m-k}dt\leq\frac{{C}}{\sigma^{k}}\omega^{m-k}_{j,p}\left(1/\omega,\>\Theta_{j}^{k}f\right),$

where

$C=\int_{-\infty}^{\infty}\rho(t)|t|^{k}(1+|t|)^{m-k}dt,$

is finite by the choice of $\rho$. The inequalities (6.8) and (6.9) are
proved.

∎

We define the following functional

(6.12) $\mathcal{E}_{p}(\sigma,f)=\inf_{g\in{\bf
B}_{p}^{\sigma}(\Theta_{1},...,\Theta_{n})}\|f-g\|_{X^{p}},\>\>1\leq
p<\infty.$

Since the operator $\mathcal{Q}_{j}(\sigma,m),\>1\leq j\leq n,$ maps
$X^{p},\>\>1\leq p<\infty,$ into ${\bf B}_{\sigma}^{p}(\Theta_{j}),\>\>1\leq
p<\infty,$ and since all the operators $\Theta_{j},\>U_{j},\>1\leq j\leq n,$
commute with each other one concludes that for any $f\in X^{p}$ the function

$\mathcal{Q}_{1}(\sigma,m)...\mathcal{Q}_{n}(\sigma,m)(f)=$
$\int_{-\infty}^{\infty}...\int_{-\infty}^{\infty}\Phi(\tau_{1})...\Phi(\tau_{n})U_{j}(\tau_{1}+...+\tau_{n})fdt,$

belongs to ${\bf B}_{\sigma}^{p}(\Theta_{1},...,\Theta_{n})$. By using (6),
the formal identity

$1-a_{1}a_{2}...a_{n}=(1-a_{a})+a_{1}(1-a_{2})+...+a_{1}a_{2}...a_{n-1}(1-a_{n}),$

and boundness of every operator $\mathcal{Q}_{j}(\sigma,m)$, we obtain the
following Jackson-type inequality

$\mathcal{E}_{p}(\sigma,f)\leq\|f-Q_{1}Q_{2}...Q_{n}f\|_{X^{p}}\leq
C\sum_{j=1}^{n}\|f-Q_{j}(\sigma,m)f\|_{X^{p}}\leq$ (6.13)
$C\sum_{j=1}^{n}\omega^{m}_{j,p}\left(1/\sigma,\>f\right)\leq
C\Omega_{p}^{m}(1/\sigma,f).$

Set

$B=\cup_{\sigma>0}{\bf B}_{\sigma}^{p}(\Theta_{1},...,\Theta_{n}),\>\>1\leq
p<\infty.$

According to the general theory (see Appendix 1) the space of all $f\in X^{p}$
for which the integral

$\left(\int_{0}^{\infty}\left(\tau^{\alpha}\mathcal{E}_{p}(\tau,f)\right)^{q}\frac{d\tau}{\tau}\right)^{1/q},\>\>1\leq
p<\infty,\>\>1\leq q\leq\infty,$

is finite is known as the approximation space and denoted by
$\mathcal{E}_{\alpha,\>q}(X^{p},B)$. We have the following Direct
Approximation Theorem.

######  Theorem 6.4.

The following continuous embedding holds $,\>\>1\leq p<\infty,\>\>1\leq
q\leq\infty,$

(6.14) $\mathcal{B}_{p,q}^{\alpha}\subset\mathcal{E}_{\alpha,\>q}(X^{p},B),$

and for some $C>0$ the inequalities hold

$\int_{0}^{\infty}\left(\tau^{\alpha}\mathcal{E}_{p}(\tau,f)\right)^{q}\frac{d\tau}{\tau}\leq
C\sum_{j=1}^{n}\int_{0}^{\infty}(s^{-\alpha}\omega^{r}_{j,p}(s,f))^{q}\frac{ds}{s}\leq
C\int_{0}^{\infty}(s^{-\alpha}\Omega^{r}_{p}(s,f))^{q}\frac{ds}{s}.$

###### Proof.

According to (6) we have

$s^{-\alpha}\mathcal{E}_{p}(s^{-1},f)\leq s^{-\alpha}\Omega_{p}^{r}(s,f).$

Since

$\int_{0}^{\infty}(s^{-\alpha}\mathcal{E}_{p}(s^{-1},f))^{q}\frac{ds}{s}=\int_{0}^{\infty}\left(\tau^{\alpha}\mathcal{E}_{p}(\tau,f)\right)^{q}\frac{d\tau}{\tau},$

and since the norm of the Besov space $\mathcal{B}^{\alpha}_{p,q}$ is
equivalent to the norm

(6.15)
$\|f\|_{X^{p}}+\left(\int_{0}^{\infty}(s^{-\alpha}\Omega^{r}_{p}(s,f))^{q}\frac{ds}{s}\right)^{1/q},\>\>\>\alpha<r,$

our claim follows from (6). Theorem is proven. ∎

In order to obtain an Inverse Approximation theorem, we are using Theorem 11.2
with ${\bf E}=X^{p},\>{\bf F}=\mathcal{W}_{p}^{m},\>\mathcal{T}=B$. On the
linear space $B=\cup_{\sigma>0}{\bf
B}_{\sigma}^{p}(\Theta_{1},...,\Theta_{n})$ we consider the quasi-norm defined
as

$|f|_{B}=\inf\left\\{\omega^{\prime}>0~{}:f\in{\bf
B}_{\sigma}^{p}(\Theta_{1},...,\Theta_{n})\right\\}~{}.$

Thus the assumption of Theorem 11.2 simply means that the Bernstein-Mellin
inequality for functions in $B$ holds true, i.e. if $f$ in $B$ belongs to
${\bf B}_{p}^{\sigma}(\Theta_{1},...,\Theta_{n})$ then

$\|f\|_{\mathcal{W}_{p}^{m}}\leq\sigma^{m}\|f\|_{X^{p}}.$

It implies the Inverse Approximation Theorem meaning that the following
continuous imbedding holds true

$\mathcal{E}_{\alpha,\>q}(X^{p},B)\subset\mathcal{B}_{p,q}^{\alpha},\>\>1\leq
p<\infty,\>\>1\leq q\leq\infty,\>\>\alpha\in\mathbb{R}_{+}.$

######  Remark 6.5.

We obtain the Direct Approximation Theorem 6.4 directly as a consequence of
the Jackson-type inequality (6). However, one could also use the abstract
Direct Theorem 11.1 to obtain Theorem 6.4. Indeed, first we note that by using
(6), (6) and (6.1) one has

$\mathcal{E}_{p}(\sigma,f)\leq
C\sum_{j=1}^{n}\|f-Q_{j}(\sigma,m)f\|_{X^{p}}\leq
C\sum_{j=1}^{n}\int_{-\infty}^{\infty}\rho(t)\omega^{m}_{j,p}\left(1/\sigma,\>f\right)\leq$
$\sum_{j=1}^{n}\frac{\omega^{m-k}_{j,p}\left(1/\sigma,\>\Theta_{j}^{k}f\right)}{\sigma^{k}}\int_{-\infty}^{\infty}\rho(t)|t|^{k}(1+|t|)^{m-k}dt\leq\sum_{j=1}^{n}\frac{{C}}{\sigma^{k}}\omega^{m-k}_{j,p}\left(1/\omega,\>\Theta_{j}^{k}f\right),$

and for $m=k$ the last inequality implies that

(6.16) $\mathcal{E}_{p}(\sigma,f)\leq
C\sigma^{-m}\|f\|_{\mathcal{W}_{p}^{m}},\>\>\>f\in\mathcal{W}_{p}^{m}.$

Comparing our situation with Theorem 11.1 we see that ${\bf E}=X^{p},\>{\bf
F}=\mathcal{W}_{p}^{m},\>\mathcal{T}=B$, and the assumption of this theorem is
exactly (6.16) with the $\beta=m$. All together it gives the imbedding (6.14).

##  7\. The Laplace-Mellin operator in the Hilbert space $X^{2}$

Since $\Theta_{j}=x_{j}\partial_{j}$ is a skew-symmetric operator in $X^{2}$
its negative square

$\Theta_{j}^{2}=-\left(x_{j}\partial_{j}\right)^{2}=-x^{2}_{j}\partial_{j}^{2}-x_{j}\partial_{j}.$

is a self-adjoint non-negative operator. We will use notation
$e^{it\Theta_{j}^{2}}$ for the corresponding group of unitary operators. In
the Hilbert space $\mathcal{W}^{k}_{2}$ we introduce the Laplace-Mellin
operator as

(7.1)
$L_{\mathcal{M}}=-\sum_{j=1}^{n}\Theta_{j}^{2}=-\sum_{j=1}^{n}\left(x^{2}_{j}\partial_{j}^{2}+x_{j}\partial_{j}\right).$

Due to commutativity of the operator $\Theta_{j}^{2}$ the self-adjoint
operator $L$ generates a group of unitary operators $e^{itL}$ and the
following formula holds

$e^{itL_{\mathcal{M}}}=e^{it\Theta_{1}^{2}}e^{it\Theta_{2}^{2}}...e^{it\Theta_{n}^{2}}.$

The operator $L_{\mathcal{M}}$ is not negative and it has a unique self-
adjoint non-negative square root $L_{\mathcal{M}}^{1/2}$. One can introduce
another scale of Sobolev-Mellin spaces which are domains
$\mathcal{D}(L_{\mathcal{M}}^{k/2}),\>k\in\mathbb{N},$ of powers of
$L_{\mathcal{M}}^{1/2}$ with the graph norm:

$|\|f\|_{\mathcal{D}(L_{\mathcal{M}}^{k/2})}=\|f\|_{X^{2}}+\|L_{\mathcal{M}}^{k/2}f\|_{X^{2}}.$

######  Theorem 7.1.

The spaces $\mathcal{D}(L_{\mathcal{M}}^{k/2})$ and
$\mathcal{W}_{2}^{k},\>k\in\mathbb{N},$ coincide and their norms are
equivalent.

###### Proof.

One has

$\sum_{j=1}^{n}\|Q_{j}f\|_{X^{2}}^{2}=\sum_{j=1}^{n}\langle\Theta_{j}f,\Theta_{j}f\rangle_{X^{2}}=\left\langle-\sum_{j=1}^{n}\Theta_{j}^{2}f,f\right\rangle_{X^{2}}=\left\langle
L_{\mathcal{M}}f,f\right\rangle_{X^{2}}=\|L_{\mathcal{M}}^{1/2}f\|_{X^{2}}.$

Thus the spaces $\mathcal{D}(L_{\mathcal{M}}^{1/2})$ and $\mathcal{W}_{2}^{1}$
coincide and their norms equivalent. Now, since $\Theta_{j}$ are commuting
with each other, we have

$\sum_{1\leq k,j\leq
n}\|\Theta_{k}\Theta_{j}f\|^{2}_{X^{2}}=\sum_{j=1}^{n}\|L_{\mathcal{M}}^{1/2}\Theta_{j}f\|^{2}=\sum_{j=1}^{n}\|\Theta_{j}L_{\mathcal{M}}^{1/2}f\|^{2}=\|L_{\mathcal{M}}f\|^{2}_{X^{2}},$

which proves the statement for $k=2$. Continue this way we can finish the
proof. ∎

We set

${\bf w}^{r}_{2}(s,f;L_{\mathcal{M}})=\sup_{0\leq\tau\leq s}\|\left(e^{i\tau
L_{\mathcal{M}}}-I\right)^{r}f\|_{X^{2}}.$

The previous Theorem implies the following Corollary.

######  Corollary 7.1.

The norm of the Besov-Mellin space $\mathcal{B}_{2,q}^{\alpha},\>1\leq
q\leq\infty,$ is equivalent to

(7.2) $\|f\|_{X^{2}}+\left(\int_{0}^{\infty}(s^{-\alpha}{\bf
w}^{r}_{2}(s,f;L_{\mathcal{M}}))^{q}\frac{ds}{s}\right)^{1/q},$

Moreover, for $\alpha$ not-integer it is equivalent to

(7.3)
$\|f\|_{\mathcal{D}\left(L_{\mathcal{M}}^{[\alpha]}\right)}+\left(\int_{0}^{\infty}\left(s^{[\alpha]-\alpha}{\bf
w}^{1}_{2}(s,L_{\mathcal{M}}^{[\alpha]}f;L_{\mathcal{M}})\right)^{q}\frac{ds}{s}\right)^{1/q}$

where $[\alpha]$ is the integer part of $\alpha$, and for
$\alpha=k\in\mathbb{N}$ integer its norm is equivalent to the norm (Zygmund
condition)

(7.4)
$\|f\|_{\mathcal{D}\left(L_{\mathcal{M}}^{k-1}\right)}+\left(\int_{0}^{\infty}\left(s^{-1}{\bf
w}^{2}_{2}(s,L^{k-1}f;L_{\mathcal{M}})\right)^{q}\frac{ds}{s}\right)^{1/q}.$

Note that according to the general theory (see for example [21], [39]) one has
the following isomorphisms for the self-adjoint operator $L^{1/2}$ for
$0<\alpha<r$:

$\mathcal{B}_{2,2}^{\alpha}=\left(X^{2},\mathcal{W}_{2}^{\alpha}\right)_{\alpha/r,\>2}^{K}=\left(X^{2},L_{\mathcal{M}}^{r/2}\right)^{K}_{\alpha/r,\>2}=\mathcal{D}\left(L_{\mathcal{M}}^{\alpha/2}\right).$

Thus when $q=2$ the norms (7.2)-(7.4) are equivalent to the graph norm of a
corresponding space $\mathcal{D}\left(L_{\mathcal{M}}^{\alpha/2}\right)$.

Let
$\mathcal{D}^{\infty}(L_{\mathcal{M}}^{1/2})=\cap_{m\in\mathbb{N}}\mathcal{D}(L_{\mathcal{M}}^{m/2})$,
where $\mathcal{D}(L_{\mathcal{M}}^{m/2})$ is the domain of
$L_{\mathcal{M}}^{m/2}.$ One has the following version of the Kolmogorov-
Stein-Mellin inequality which follows from Theorem 12.3.

######  Theorem 7.2.

The following holds for $f\in\mathcal{D}(L_{\mathcal{M}}^{m/2})$

(7.5) $\left\|L_{\mathcal{M}}^{k/2}f\right\|_{X^{2}}^{m}\leq
C_{k,m}\|L_{\mathcal{M}}^{m/2}f\|_{X^{2}}^{k}\|f\|_{X^{2}}^{m-k},\>\>\>0\leq
k\leq m,$

where $C_{k,m}=(K_{m-k})^{m}/(K_{m})^{m-k},$ and $K_{j}$ is the Favard
constant.

Consider the following inequality

(7.6) $\|L_{\mathcal{M}}^{k/2}f\|_{X^{2}}\leq\sigma^{k}\|f\|_{X^{2}},$

and let $\mathbf{B}^{2}_{\sigma}(L_{\mathcal{M}}^{1/2}),$ be the space of all
$f\in X^{2}$ for which this inequality holds. For a function
$f\in\cup_{\sigma>0}\mathbf{B}^{2}_{\sigma}(L_{\mathcal{M}}^{1/2}),$ let the
$\sigma_{f}$ be the smallest $\sigma$ for which the inequality (7.6) holds.

The next Theorem follows from Theorem 12.4 in Appendix 2.

######  Theorem 7.3.

Let $f\in X^{2}$ belongs to a space
$\mathbf{B}^{2}_{\sigma}(L_{\mathcal{M}}^{1/2}),$ for some $0<\sigma<\infty.$
Then the following limit exists

(7.7)
$d_{j;f}=\lim_{k\rightarrow\infty}\|L_{\mathcal{M}}^{k/2}f\|^{1/k}_{X^{2}},$

and $d_{j,f}=\sigma_{j,f}.$ Conversely, if
$f\in\mathcal{D}^{\infty}(L_{\mathcal{M}}^{1/2})$ and
$d_{j,f}=\lim_{k\rightarrow\infty}\|L_{\mathcal{M}}^{k/2}f\|^{1/k}_{X^{2}},$
exists and is finite, then
$f\in\mathbf{B}^{2}_{d_{j,f}}(L_{\mathcal{M}}^{1/2})$ and
$d_{j,f}=\sigma_{j,f}.$

##  8\. Paley-Wiener-Mellin spaces $PW_{\sigma}(L_{\mathcal{M}}^{1/2})$

Consider a self-adjoint non-negative definite operator $L_{\mathcal{M}}$ in
the Hilbert space $X^{2}$ and let $\sqrt{L_{\mathcal{M}}}$ be the non-negative
square root of $L_{\mathcal{M}}$.

According to the spectral theory for self-adjoint operators [13] there exists
a direct integral of Hilbert spaces $H=\int H(\lambda)dm(\lambda)$ and a
unitary operator $\mathcal{F}$ from $X^{2}$ onto $H$, which transforms the
domains of $L_{\mathcal{M}}^{k/2},k\in\mathbb{N},$ onto the sets
$H_{k}=\\{x\in H|\lambda^{k}x\in H\\}$ with the norm

(8.1)
$\|x(\lambda)\|_{H_{k}}=\left<x(\lambda),x(\lambda)\right>^{1/2}_{H(\lambda)}=\left(\int^{\infty}_{0}\lambda^{2k}\|x(\lambda)\|^{2}_{H(\lambda)}dm(\lambda)\right)^{1/2}.$

and satisfies the identity
$\mathcal{F}(L_{\mathcal{M}}^{k/2}f)(\lambda)=\lambda^{k}(\mathcal{F}f)(\lambda),$
if $f$ belongs to the domain of $L_{\mathcal{M}}^{k/2}$. We call the operator
$\mathcal{F}$ the Spectral Fourier Transform [33]. As known, $H$ is the set of
all $m$-measurable functions $\lambda\mapsto x(\lambda)\in H(\lambda)$, for
which the following norm is finite:

$\|x\|_{H}=\left(\int^{\infty}_{0}\|x(\lambda)\|^{2}_{H(\lambda)}dm(\lambda)\right)^{1/2}$

For a function $F$ on $[0,\infty)$ which is bounded and measurable with
respect to $dm$ one can introduce the operator $F(L_{\mathcal{M}})$ by using
the formula

(8.2) $F(L_{\mathcal{M}})f=\mathcal{F}^{-1}F(\lambda)\mathcal{F}f,\>\>\>f\in
X^{2}.$

If $F$ is real-valued the operator $F(L_{\mathcal{M}})$ is self-adjoint.

######  Definition 4.

We say that a function $f\in X^{2}$ belongs to the Paley-Wiener-Mellin space
$PW_{\sigma}\left(L_{\mathcal{M}}^{1/2}\right)$ if the support of the Spectral
Fourier Transform $\mathcal{F}f$ is contained in $[0,\sigma]$.

The next two facts are obvious.

######  Theorem 8.1.

The spaces $PW_{\sigma}\left(L_{\mathcal{M}}^{1/2}\right)$ have the following
properties:

  1. (1)

the space $PW_{\sigma}\left(L_{\mathcal{M}}^{1/2}\right)$ is a linear closed
subspace in $X^{2}$,

  2. (2)

the space $\bigcup_{\sigma>0}PW_{\sigma}\left(L_{\mathcal{M}}^{1/2}\right)$ is
dense in $X^{2}$.

The next theorem contains generalizations of several results from classical
harmonic analysis (in particular the Paley-Wiener theorem). It follows from
our results in [33].

######  Theorem 8.2.

The following statements hold:

  1. (1)

(Bernstein-Mellin inequality) $f\in
PW_{\sigma}\left(L_{\mathcal{M}}^{1/2}\right)$ if and only if
$f\in\mathcal{D}^{\infty}(L_{\mathcal{M}})=\bigcap_{k=1}^{\infty}\mathcal{D}(L_{\mathcal{M}}^{k})$,
and the following Bernstein-type inequalities holds true

(8.3)
$\|L_{\mathcal{M}}^{s/2}f\|_{X^{2}}\leq\sigma^{s}\|f\|_{X^{2}}\quad\mbox{for
all}\,\,s\in\mathbb{R}_{+};$

  2. (2)

(Paley-Wiener-Mellin theorem) $f\in PW_{\sigma}(L_{\mathcal{M}}^{1/2})$ if and
only if for every $g\in X^{2}$ the scalar-valued function of the real variable
$t\mapsto\langle e^{it\sqrt{L_{\mathcal{M}}}}f,g\rangle$ is bounded on the
real line and has an extension to the complex plane as an entire function of
the exponential type $\omega$;

  3. (3)

(Riesz-Boas-Mellin interpolation formula) $f\in
PW_{\sigma}\left(L_{\mathcal{M}}^{1/2}\right)$ if and only if
$f\in\mathcal{D}(L_{\mathcal{M}}^{\infty})$ and the following Riesz-Boas-
Mellin interpolation formula holds for all $\sigma>0$:

(8.4)
$i\sqrt{L_{\mathcal{M}}}f=\frac{\sigma}{\pi^{2}}\sum_{k\in\mathbb{Z}}\frac{(-1)^{k-1}}{(k-1/2)^{2}}e^{i\left(\frac{\pi}{\omega}(k-1/2)\right)\sqrt{L_{\mathcal{M}}}}f.$

###### Proof.

(1) follows immediately from the definition and representation (8.1). To prove
(2) it is sufficient to apply the classical Bernstein inequality [24] in the
uniform norm on $\mathbb{R}$ to every function $\langle
e^{it\sqrt{L_{\mathcal{M}}}}f,g\rangle,\>\>g\in X^{2}$. To prove (3) one has
to apply the classical Riesz-Boas interpolation formula on $\mathbb{R}$ [24],
[35], to the function $\langle e^{it\sqrt{L_{\mathcal{M}}}}f,g\rangle$. ∎

The Bernstein inequality (8.3) and Theorem 7.1 imply the following.

######  Theorem 8.3.

The Bernstein-Mellin space ${\bf B}_{2}^{\sigma}(\Theta_{1},...,\Theta_{n})$
and the Paley-Wiener-Mellin space $PW_{\sigma}(L_{\mathcal{M}}^{1/2})$
coincide.

Thus the best approximation functional $\mathcal{E}_{2}(s,f)$ defined in
(6.12) can be redefined as

$\mathcal{E}_{2}(\sigma,f)=\mathcal{E}_{2}(\sigma,f;L_{\mathcal{M}})=\inf_{g\in
PW_{\sigma}(L_{\mathcal{M}}^{1/2})}\|f-g\|_{X^{2}}.$

For a different approach to the Paley-Wiener-Mellin spaces in terms of the
Mellin transform (1.3) and its inverse (1.4) we refer to the very interesting
papers [7], [8], [10].

##  9\. Paley-Wiener-Mellin frames in $X^{2}$.

###  9.1. Partitions of unity on the frequency side

The construction of frequency-localized frames is achieved via spectral
calculus. The idea is to start from a partition of unity on the positive real
axis. In the following, we will be considering two different types of such
partitions, whose construction we now describe in some detail. The
construction below was discribed in [19].

Let $g\in C^{\infty}(\mathbb{R}_{+})$ be a non-increasing function such that
$supp(g)\subset[0,\>2],$ and $g(\lambda)=1$ for $\lambda\in[0,\>1],\>0\leq
g(\lambda)\leq 1,\>\lambda>0.$ We now let
$h(\lambda)=g(\lambda)-g(2\lambda)~{},$ which entails
$supp(h)\subset[2^{-1},2]$, and use this to define

$F_{0}(\lambda)=\sqrt{g(\lambda)}~{},F_{j}(\lambda)=\sqrt{h(2^{-j}\lambda)}~{},j\geq
1~{},$

as well as
$G_{j}(\lambda)=\left[F_{j}(\lambda)\right]^{2}=F_{j}^{2}(\lambda)~{},j\geq
0~{}.$ As a result of the definitions, we get for all $\lambda\geq 0$ the
equations

$\sum_{j\in\mathbb{N}}G_{j}(\lambda)=\sum_{j\in\mathbb{N}}F_{j}^{2}(\lambda)=g(2^{-n}\lambda),$

and as a consequence

$\sum_{j\in\mathbb{N}}G_{j}(\lambda)=\sum_{j\in\mathbb{N}}F_{j}^{2}(\lambda)=1~{},\>\>\>\lambda\geq
0,$

with finitely many nonzero terms occurring in the sums for each fixed
$\lambda$. We call the sequence $(G_{j})_{j\geq 0}$ a (dyadic) partition of
unity, and $(F_{j})_{j\geq 0}$ a quadratic (dyadic) partition of unity. As
will become soon apparent, quadratic partitions are useful for the
construction of frames. Using the spectral theorem one has

$F_{j}^{2}(L_{\mathcal{M}})f=\mathcal{F}^{-1}\left(F_{j}^{2}(\lambda)\mathcal{F}f(\lambda)\right),\>\>\>j\geq
1,$

and thus

(9.1)
$f=\mathcal{F}^{-1}\mathcal{F}f(\lambda)=\mathcal{F}^{-1}\left(\sum_{j\in\mathbb{N}}F_{j}^{2}(\lambda)\mathcal{F}f(\lambda)\right)=\sum_{j\in\mathbb{N}}F_{j}^{2}(L_{\mathcal{M}})f$

Taking inner product with $f$ gives

$\|F_{j}(L_{\mathcal{M}})f\|^{2}_{X^{2}}=\langle
F_{j}^{2}(L_{\mathcal{M}})f,f\rangle,$

and

(9.2) $\|f\|_{X^{2}}^{2}=\sum_{j\in\mathbb{N}}\langle
F_{j}^{2}(L_{\mathcal{M}})f,f\rangle=\sum_{j\in\mathbb{N}}\|F_{j}(L_{\mathcal{M}})f\|_{X^{2}}^{2}.$

Similarly, we get the identity
$\sum_{j\in\mathbb{N}}G_{j}(L_{\mathcal{M}})f=f~{}.$ Moreover, since the
functions $G_{j},F_{j}$, have their supports in $[2^{j-1},\>\>2^{j+1}]$, the
elements $F_{j}(L_{\mathcal{M}})f$ and $G_{j}(L_{\mathcal{M}})f$ are
bandlimited to $[2^{j-1},\>\>2^{j+1}]$, whenever $j\geq 1$, and to $[0,2]$ for
$j=0$.

###  9.2. Paley-Wiener-Mellin frames in $X^{2}$

We consider the Laplace-Mellin operator $L(L_{\mathcal{M}})$ defined in (7.1)
in the Hilbert spaces $X^{2}$.

######  Definition 5.

For every $j\in\mathbb{N}$ let

$\\{\Phi_{k}^{j}\\}_{k=1}^{K_{j}},\>\>\>\>\>\>\Phi_{k}^{j}\in
PW_{[2^{j-1},\>2^{j+1})}\left(L_{\mathcal{M}}^{1/2}\right),$
$K_{j}\in\mathbb{N}\cup\\{\infty\\},$

be a frame in $PW_{[2^{j-1},\>2^{j+1})}\left(L_{\mathcal{M}}^{1/2}\right)$
with the fixed constants $a,\>b$, i.e.

(9.3)
$a\|f\|_{X^{2}}^{2}\leq\sum_{k=1}^{K_{j}}\left|\left<f,\Phi^{j}_{k}\right>\right|^{2}\leq
b\|f\|_{X^{2}}^{2},\>\>\>f\in
PW_{[2^{j-1},\>2^{j+1})}\left(L_{\mathcal{M}}^{1/2}\right).$

The formula (9.2) and the general theory of frames imply the following
statement.

######  Theorem 9.1.

  1. (1)

The set of functions $\\{\Phi_{k}^{j}\\}$ will be a frame in the entire space
$X^{2}$ with the same frame constants $a$ and $b$, i.e.

(9.4)
$a\|f\|_{X^{2}}^{2}\leq\sum_{j}\sum_{k}\left|\left<f,\Phi^{j}_{k}\right>\right|^{2}\leq
b\|f\|_{X^{2}}^{2},\>\>\>f\in X^{2}.$

  2. (2)

The canonical dual frame $\\{\Psi^{j}_{k}\\}$ also consists of bandlimited
vectors $\Psi^{j}_{k}\in
PW_{[2^{j-1},\>2^{j+1}]}(L_{\mathcal{M}}^{1/2}),\>\>j\in\mathbb{N},\>k=1,...,K_{j}$,
and has the frame bounds $b^{-1},\>\>a^{-1}.$

  3. (3)

The reconstruction formulas hold for every $f\in X^{2}$

$f=\sum_{j}\sum_{k}\left<f,\Phi^{j}_{k}\right>\Psi^{j}_{k}=\sum_{j}\sum_{k}\left<f,\Psi^{j}_{k}\right>\Phi^{j}_{k}.$

The formula (9.2) implies that in this case the set of functions
$\\{\Phi_{k}^{j}\\}$ will be a frame in the entire $X^{2}$ with the same frame
constants $a$ and $b$, i.e.

(9.5)
$a\|f\|_{X^{2}}^{2}\leq\sum_{j}\sum_{k}\left|\left<f,\Phi^{j}_{k}\right>\right|^{2}\leq
b\|f\|_{X^{2}}^{2},\>\>\>f\in X^{2}.$

##  10\. More about Besov-Mellin spaces $\mathcal{B}^{\sigma}_{2,q}$

In this section we applying Theorems 11.1 and 11.2 to a situation where ${\bf
E}=X^{2},\>{\bf
F}=\mathcal{W}_{2}^{r},\>\mathcal{B}_{2,q}^{\alpha}=(X^{2},\mathcal{W}_{2}^{r})^{K}_{\alpha/r,q},$
and $\mathcal{T}=\cup_{\omega>0}PW_{\omega}\left(L_{\mathcal{M}}^{1/2}\right)$
is the abelian additive group of the underlying vector space, with the quasi-
norm

$\|f\|_{\mathcal{T}}=\inf\left\\{\omega^{\prime}>0~{}:f\in
PW_{\mathbf{\omega}^{\prime}}\left(L_{\mathcal{M}}^{1/2}\right)\right\\}~{}.$

In previous sections we already proved that in our case the assumptions of
Theorems 11.1 and 11.2 are satisfied. It allows us to formulate the following
result.

######  Theorem 10.1.

For $\alpha>0,\>\>\>1\leq q\leq\infty,$ the norm of
$\mathcal{B}_{2,q}^{\alpha}$, is equivalent to

(10.1)
$\|f\|_{X^{2}}+\left(\sum_{j=0}^{\infty}\left(2^{j\alpha}\mathcal{E}_{2}(f,2^{j};L_{\mathcal{M}}\right)^{q}\right)^{1/q}.$

Let the functions $F_{j}$ be as in Subsection 9.1.

######  Theorem 10.2.

For $\alpha>0,\>\>\>1\leq q\leq\infty,$ the norm of
$\mathcal{B}_{2,q}^{\alpha}$, is equivalent to

(10.2)
$f\mapsto\left(\sum_{j=0}^{\infty}\left(2^{j\alpha}\left\|F_{j}(L_{\mathcal{M}})f\right\|_{X^{2}}\right)^{q}\right)^{1/q},$

with the standard modifications for $q=\infty$.

###### Proof.

We obviously have

$\mathcal{E}_{2}(f,2^{l};L_{\mathcal{M}})\leq\sum_{j>l}\left\|F_{j}(L_{\mathcal{M}})f\right\|_{X^{2}}.$

By using a discrete version of Hardy’s inequality [14] we obtain the estimate

(10.3)
$\|f\|+\left(\sum_{l=0}^{\infty}\left(2^{l\alpha}\mathcal{E}_{2}(f,2^{l};L_{\mathcal{M}})\right)^{q}\right)^{1/q}\leq
C\left(\sum_{j=0}^{\infty}\left(2^{j\alpha}\left\|F_{j}(L_{\mathcal{M}})f\right\|_{X^{2}}\right)^{q}\right)^{1/q}.$

Conversely, for any $g\in PW_{2^{j-1}}\left(L_{\mathcal{M}}^{1/2}\right)$ we
have

$\left\|F_{j}(L_{\mathcal{M}})f\right\|_{X^{2}}=\left\|F_{j}(L_{\mathcal{M}})(f-g)\right\|_{X^{2}}\leq\|f-g\|_{X^{2}}.$

This implies the estimate

$\left\|F_{j}(L_{\mathcal{M}})f\right\|_{X^{2}}\leq\mathcal{E}_{2}(f,\>2^{j-1};L_{\mathcal{M}}),$

which shows that the inequality opposite to (10.3) holds. The proof is
complete. ∎

######  Theorem 10.3.

For $\alpha>0,\>\>\>1\leq q\leq\infty,$ the norm of
$\mathcal{B}_{2,q}^{\alpha}$ is equivalent to

(10.4) $\left(\sum_{j=0}^{\infty}2^{j\alpha
q}\left(\sum_{k}\left|\left<f,\Phi^{j}_{k}\right>\right|^{2}\right)^{q/2}\right)^{1/q}\asymp\|f\|_{\mathcal{B}_{2,q}^{\alpha}},$

with the standard modifications for $q=\infty$.

###### Proof.

For $f\in X^{2}$ and operator $F_{j}(L_{\mathcal{M}})$ we have

(10.5)
$a\left\|F_{j}(L_{\mathcal{M}})f\right\|_{X^{2}}^{2}\leq\sum_{k}\left|\left<F_{j}(L_{\mathcal{M}})f,\phi^{j}_{k}\right>\right|^{2}\leq
b\left\|F_{j}(L_{\mathcal{M}})f\right\|_{X^{2}}^{2},$

and then we obtain the following inequality

$\sum_{k}\left|\left<f,\Phi^{j}_{k}\right>\right|^{2}\leq\left\|F_{j}(L_{\mathcal{M}})f\right\|_{X^{2}}^{2}\leq
a^{-1}\sum_{k}\left|\left<f,\Phi^{j}_{k}\right>\right|^{2}\quad\mbox{for
all}\,\,\,f\in X_{2}.$

Theorem is proven. ∎

##  11\. Appendix 1.

###  11.1. $K$-functional and interpolation spaces

The goal of the section is to introduce basic notions of the theory of
interpolation spaces [14], [12], [21], [39], and approximation spaces [12],
[18], [22], [25]. It is important to realize that the relations between
interpolation and approximation spaces cannot be described in the language of
normed spaces. We have to make use of quasi-normed linear spaces in order to
treat them simultaneously.

A quasi-norm $\|\cdot\|_{\bf E}$ on linear space $\bf E$ is a real-valued
function on $\bf E$ such that for any $f,f_{1},f_{2}\in{\bf E}$ the following
holds true:

  1. (1)

$\|f\|_{\bf E}\geq 0;\>\>\>$

  2. (2)

$\|f\|_{{\bf E}}=0\Longleftrightarrow f=0;\>\>\>$

  3. (3)

$\|-f\|_{{\bf E}}=\|f\|_{{\bf E}};\>\>\>$

  4. (4)

there exists some $C_{{\bf E}}\geq 1$ such that $\|f_{1}+f_{2}\|_{{\bf E}}\leq
C_{{\bf E}}(\|f_{1}\|_{{\bf E}}+\|f_{2}\|_{{\bf E}}).\>\>$

Two quasi-normed linear spaces ${\bf E}$ and ${\bf F}$ form a pair if they are
linear subspaces of a common linear space ${\bf A}$ and the conditions
$\|f_{k}-g\|_{{\bf E}}\rightarrow 0,$ and $\|f_{k}-h\|_{{\bf F}}\rightarrow 0$
imply equality $g=h$ (in ${\bf A}$). For any such pair ${\bf E},{\bf F}$ one
can construct the space ${\bf E}\cap{\bf F}$ with quasi-norm

$\|f\|_{{\bf E}\cap{\bf F}}=\max\left(\|f\|_{{\bf E}},\|f\|_{{\bf F}}\right)$

and the sum of the spaces, ${\bf E}+{\bf F}$ consisting of all sums
$f_{0}+f_{1}$ with $f_{0}\in{\bf E},f_{1}\in{\bf F}$, and endowed with the
quasi-norm

$\|f\|_{{\bf E}+{\bf F}}=\inf_{f=f_{0}+f_{1},f_{0}\in{\bf E},f_{1}\in{\bf
F}}\left(\|f_{0}\|_{{\bf E}}+\|f_{1}\|_{{\bf F}}\right).$

Quasi-normed spaces ${\bf H}$ with ${\bf E}\cap{\bf F}\subset{\bf
H}\subset{\bf E}+{\bf F}$ are called intermediate between ${\bf E}$ and ${\bf
F}$. If both $E$ and $F$ are complete the inclusion mappings are automatically
continuous. An additive homomorphism $T:{\bf E}\rightarrow{\bf F}$ is called
bounded if

$\|T\|=\sup_{f\in{\bf E},f\neq 0}\|Tf\|_{{\bf F}}/\|f\|_{{\bf E}}<\infty.$

An intermediate quasi-normed linear space ${\bf H}$ interpolates between ${\bf
E}$ and ${\bf F}$ if every bounded homomorphism $T:{\bf E}+{\bf
F}\rightarrow{\bf E}+{\bf F}$ which is a bounded homomorphism of ${\bf E}$
into ${\bf E}$ and a bounded homomorphism of ${\bf F}$ into ${\bf F}$ is also
a bounded homomorphism of ${\bf H}$ into ${\bf H}$. On ${\bf E}+{\bf F}$ one
considers the so-called Peetre’s $K$-functional

$K(f,t)=K(f,t,{\bf E},{\bf F})=\inf_{f=f_{0}+f_{1},f_{0}\in{\bf
E},f_{1}\in{\bf F}}\left(\|f_{0}\|_{{\bf E}}+t\|f_{1}\|_{{\bf F}}\right).$

The quasi-normed linear space $({\bf E},{\bf F})^{K}_{\theta,q}$, with
parameters $0<\theta<1,\,0<q\leq\infty$, or $0\leq\theta\leq 1,\,q=\infty$, is
introduced as the set of elements $f$ in ${\bf E}+{\bf F}$ for which

(11.1)
$\|f\|_{\theta,q}=\left(\int_{0}^{\infty}\left(t^{-\theta}K(f,t)\right)^{q}\frac{dt}{t}\right)^{1/q}<\infty.$

It turns out that $({\bf E},{\bf F})^{K}_{\theta,q}$ with the quasi-norm
(11.1) interpolates between ${\bf E}$ and ${\bf F}$.

Let us introduce another functional on ${\bf E}+{\bf F}$, where ${\bf E}$ and
${\bf F}$ form a pair of quasi-normed linear spaces

$\mathcal{E}(f,t)=\mathcal{E}(f,t,\mathbf{E},\mathbf{F})=\inf_{g\in{\bf
F},\|g\|_{{\bf F}}\leq t}\|f-g\|_{{\bf E}}.$

######  Definition 6.

The approximation space $\mathcal{E}_{\alpha,q}({\bf E},{\bf
F}),0<\alpha<\infty,0<q\leq\infty$ is the quasi-normed linear spaces of all
$f\in{\bf E}+{\bf F}$ for which the quasi-norm

(11.2) $\|f\|_{\mathcal{E}_{\alpha,q}({\bf E},{\bf
F})}=\left(\int_{0}^{\infty}\left(t^{\alpha}\mathcal{E}(f,t)\right)^{q}\frac{dt}{t}\right)^{1/q}$

is finite.

###  11.2. Approximation spaces

The next two theorems represent a very abstract version of what is known as a
Direct and an Inverse Approximation Theorems [25, 13]. In the form it is
stated below they were proved in [22].

######  Theorem 11.1.

Suppose that $\mathcal{T}\subset{\bf F}\subset{\bf E}$ are quasi-normed linear
spaces and ${\bf E}$ and ${\bf F}$ are complete. If there exist $C>0$ and
$\beta>0$ such that the following Jackson-type inequality is satisfied
$t^{\beta}\mathcal{E}(t,f,\mathcal{T},{\bf E})\leq C\|f\|_{{\bf
F}},\>\>t>0,\>\>f\in{\bf F},$ then the following embedding holds true

(11.3) $({\bf E},{\bf
F})^{K}_{\theta,q}\subset\mathcal{E}_{\theta\beta,q}({\bf
E},\mathcal{T}),\quad\>0<\theta<1,\>0<q\leq\infty.$

######  Theorem 11.2.

If there exist $C>0$ and $\beta>0$ such that the following Bernstein-type
inequality holds $\|f\|_{{\bf F}}\leq C\|f\|^{\beta}_{\mathcal{T}}\|f\|_{{\bf
E}},\>\>f\in\mathcal{T},$ then the following embedding holds true

(11.4) $\mathcal{E}_{\theta\beta,q}({\bf E},\mathcal{T})\subset({\bf E},{\bf
F})^{K}_{\theta,q},\quad 0<\theta<1,\>0<q\leq\infty.$

###  11.3. Frames in Hilbert spaces

A family of vectors $\\{\Phi_{j}\\}$ in a Hilbert space $\mathbf{H}$ is called
a frame if there exist constants $A,B>0$ such that

(11.5)
$A\|f\|^{2}_{\mathbf{H}}\leq\sum_{j}\left|\left<f,\Phi_{j}\right>\right|^{2}\leq
B\|f\|_{\mathbf{H}}^{2}\quad\mbox{for all}\quad f\in\mathbf{H}.$

The largest $A$ and smallest $B$ are called lower and upper frame bounds.

The family of scalars $\\{\left<f,\Phi_{j}\right>\\}$ represents a set of
measurements of a vector $f$. In order to reconstruct the vector $f$ from this
collection of measurements in a linear way one has to find another (dual)
frame $\\{\Psi_{j}\\}$. Then a reconstruction formula is

$f=\sum_{j}\left<f,\Phi_{j}\right>\Psi_{j}.$

Dual frames are not unique in general. Moreover it may be difficult to find a
dual frame in concrete situations. If $A=B=1$ the frame is said to be tight or
Parseval. Parseval frames are similar in many respects to orthonormal wavelet
bases. For example, if in addition all vectors $\Phi_{j}$ are unit vectors,
then the frame is an orthonormal basis. The main feature of Parseval frames is
that decomposing and synthesizing a vector from known data are tasks carried
out with the same family of functions, i.e., the Parseval frame is its own
dual frame.

For more details on Interpolation and Approximation spaces see [14], [12],
[18], [21], [22], [23], [25], [39].

##  12\. Appendix 2.

Let $D$ be a generator of a strongly continuous one parameter group
$T(t),\>t\in\mathbb{R},$ of isometries of a Banach space $\mathbb{B}$ with the
norm $\|\cdot\|$. Let
$\mathcal{D}^{\infty}=\cap_{m\in\mathbb{N}}\mathcal{D}(D^{m})$, where
$\mathcal{D}(D^{m})$ is the domain of $D^{m}.$ Let $\mathbb{B}_{\sigma}(D)$ be
the space of all $f\in\mathbb{B}$ for which the Bernstein-type inequality
holds, i.e.

(12.1) $\|D^{k}f\|\leq\sigma^{k}\|f\|,\>k\in\mathbb{N}.$

The following statements [33], [35], imply Lemma 5.1 and Theorem 6.1
respectively.

######  Lemma 12.1.

A function $f\in\mathcal{D}^{\infty}(D)$ belongs to
$\mathbb{B}_{\sigma}(D),\>1\leq j\leq n,>\>\sigma>0,$ if and only if the
quantity

(12.2) $\sup_{k\in N}\sigma^{-k}\|D^{k}f\|=R(f,\sigma)$

is finite.

######  Theorem 12.2.

Suppose that $p\in{\bf B}^{1}_{\sigma}(\mathbb{R})$ is an entire function of
exponential type $\sigma$. Then for any $f\in\mathbb{B}$ the function

$P_{j,\sigma}f=\int_{-\infty}^{\infty}p(t)T(t)fdt$

belongs to $\mathbb{B}_{\sigma}^{p}(D).$ The integral here is understood in
the sense of Bocher for the abstract-valued function
$p(t)T(t)f:\mathbb{R}\mapsto\mathbb{B}$.

###  12.1. Kolmogorov-Stein inequality

Let us introduce the Favard constants (see [1], Ch. V) which are defined as

$K_{j}=\frac{4}{\pi}\sum_{r=0}^{\infty}\frac{(-1)^{r(j+1)}}{(2r+1)^{j+1}},\>\>\>j,\>\>r\in\mathbb{N}.$

It is known, that the sequence of all Favard constants with even indices is
strictly increasing and belongs to the interval $[1,4/\pi)$ and the sequence
of all Favard constants with odd indices is strictly decreasing and belongs to
the interval $(\pi/4,\pi/2],$ i.e.,

(12.3) $K_{2j}\in[1,4/\pi),\;K_{2j+1}\in(\pi/4,\pi/2].$

We will need a generalization of the classical Kolmogorov-Stein inequality.
This inequality was first proved by Kolmogorov for $L^{\infty}(\mathbb{R})$
and later extended to $L^{p}(\mathbb{R})$ for $1\leq p<\infty$ by Stein [38].

######  Lemma 12.3.

If $f\in\mathcal{D}(D^{m})$ then the following inequality holds

(12.4) $\left\|D^{k}f\right\|^{m}\leq
C_{k,m}\|D^{m}f\|^{k}\|f\|^{m-k},\>\>\>0\leq k\leq m,$

where $C_{k,m}=(K_{m-k})^{m}/(K_{m})^{m-k}.$

###### Proof.

Indeed, for any $h^{*}\in\mathbb{B}^{*}$ the Kolmogorov inequality [38]
applied to the entire function $\left<T(t)f,h^{*}\right>$ gives

$\left\|\left(\frac{d}{dt}\right)^{k}\left<T(t)f,h^{*}\right>\right\|^{m}_{C(\mathbb{R}^{1})}\leq
C_{k,m}\left\|\left(\frac{d}{dt}\right)^{n}\left<T(t)f,h^{*}\right>\right\|_{C(\mathbb{R}^{1})}^{k}\times$
$\left\|\left<T(t)f,h^{*}\right>\right\|_{C(\mathbb{R}^{1})}^{m-k},\quad
0<k<m,$

or

$\left\|\left<T(t)D^{k}f,h^{*}\right>\right\|_{C(\mathbb{R}^{1})}^{m}\leq
C_{k,m}\left\|\left<T(t)D^{m}f,h^{*}\right>\right\|_{C(\mathbb{R}^{1})}^{k}\left\|\left<T(t)f,h^{*}\right>\right\|_{C(\mathbb{R}^{1})}^{m-k}.$

We obtain

$\displaystyle\left\|\left<T(t)D^{k}f,h^{*}\right>\right\|_{C(\mathbb{R}^{1})}^{m}$
$\displaystyle\leq
C_{k,m}\|h^{*}\|^{k}\|D^{m}f\|^{k}\|h^{*}\|^{m-k}\|f\|^{m-k}$
$\displaystyle\leq C_{k,m}\|h^{*}\|^{m}\|D^{m}f\|^{k}\|f\|^{m-k},$

which yields when $t=0$

$\left|\left<D^{k}f,h^{*}\right>\right|^{m}\leq
C_{k,m}\|h^{*}\|^{m}\|D^{m}f\|^{k}\|f\|^{m-k}.$

By choosing $h$ such that $\left|<D^{k}f,h^{*}>\right|=\|D^{k}f\|$ and
$\|h^{*}\|=1$ we obtain (12.4). ∎

###  12.2. A characterization of $\mathbb{B}_{\sigma}(D)$

For a vector $f\in\cup_{\sigma>0}\mathbb{B}_{\sigma}(D),$ the $\sigma_{f}$
will denote the smalest $\sigma$ for which the inequality (12.1) holds.

######  Theorem 12.4.

Let $f\in\mathbb{B}$ belongs to a space $\mathbb{B}_{\sigma}(D),$ for some
$0<\sigma<\infty.$ Then the following limit exists

(12.5) $d_{f}=\lim_{k\rightarrow\infty}\|D^{k}f\|^{1/k},$

and $d_{f}=\sigma_{f}.$ Conversely, if $f\in\mathcal{D}^{\infty}$ and
$d_{f}=\lim_{k\rightarrow\infty}\|D^{k}f\|^{1/k},$ exists and is finite, then
$f\in\mathbb{B}_{d_{f}}(D)$ and $d_{f}=\sigma_{f}.$

###### Proof.

According to Lemma 12.3 we have

$\left\|D^{k}f\right\|^{n}\leq C_{k,m}\|D^{m}f\|^{k}\|f\|^{m-k},\quad 0\leq
k\leq m.$

Without loss of generality, let us assume that $\|f\|=1.$ Thus,

$\left\|D^{k}f\right\|^{1/k}\leq(\pi/2)^{1/km}\|D^{m}f\|^{1/m},\quad 0\leq
k\leq m.$

Let $k$ be arbitrary but fixed. It follows that

$\left\|D^{k}f\right\|^{1/k}\leq(\pi/2)^{1/km}\|D^{m}f\|^{1/m},\mbox{ for all
}m\geq k,$

which implies that

$\left\|D^{k}f\right\|^{1/k}\leq\underline{\lim}_{n\rightarrow\infty}\|D^{m}f\|^{1/m}.$

But since this inequality is true for all $k>0,$ we obtain that

$\overline{\lim}_{k\rightarrow\infty}\|D^{k}f\|^{1/k}\leq\underline{\lim}_{m\rightarrow\infty}\|D^{m}f\|^{1/m},$

which proves that $d_{f}=\lim_{k\rightarrow}\|D^{k}f\|^{1/k}$ exists. Since
$f\in\mathbb{B}_{\sigma}(D)$ the constant $\sigma_{f}$ is finite and we have

$\|D^{k}f\|^{1/k}\leq\sigma_{f}\|f\|^{1/k},$

and by taking the limit as $k\rightarrow\infty$ we obtain
$d_{f}\leq\sigma_{f}.$ To show that $d_{f}=\sigma_{f},$ let us assume that
$d_{f}<\sigma_{f}.$ Therefore, there exist $M>0$ and $\sigma$ such that
$0<d_{f}<\sigma<\sigma_{f}$ and

$\|D^{k}f\|\leq M\sigma^{k},\quad\mbox{for all }k>0.$

Thus, by Lemma 5.4 we have $f\in\mathbb{B}_{\sigma}(D),$ which is a
contradiction to the definition of $\sigma_{f}.$

Conversely, suppose that $d_{f}=\lim_{k\rightarrow\infty}\|D^{k}f\|^{1/k}$
exists and is finite. Therefore, there exist $M>0$ and $\sigma>0$ such that
$d_{f}<\sigma$ and

$\|D^{k}f\|\leq M\sigma^{k},\quad\mbox{for all }k>0,$

which, in view of Lemma 5.4, implies that $f\in\mathbb{B}_{\sigma}(D).$ Now by
repeating the argument in the first part of the proof we obtain
$d_{f}=\sigma_{f},$ where
$\sigma_{f}=\inf\left\\{\sigma:f\in\mathbb{B}_{\sigma}(D)\right\\}.$

Theorem 7.3 is proven. ∎

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