\documentclass[12pt, reqno]{amsart}
\usepackage{hyperref,amssymb}
\usepackage{a4wide}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{cite}
%\usepackage[numbers,sort&compress]{natbib}
\usepackage{amsmath}
\usepackage{geometry}
\geometry{margin=1in}

\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}{Corollary}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{prop}{Proposition}[section]
\theoremstyle{definition}
\newtheorem{dfn}{Definition}[section]
\newtheorem{remark}{Remark}
\newtheorem{example}{Example}
\numberwithin{equation}{section}

\title[Approximations in $l_p$ and Morrey sequence spaces]{Approximations in $l_p$ and Morrey sequence spaces}
\author{Nickolai Kudryavtsev}
\address{Faculty of Mechanics \& Mathematics, Moscow State University, 1 Leninskiye Gory, Main Building, 119991, Moscow, GSP-1, Russia}
\email{nick.kudryav@gmail.com, nilkud@mech.math.msu.su}

\begin{document}

\subjclass{40A25, 41A17, 46A45, 46E39}

\keywords{Approximations, differences, $l_p$ spaces, Besov sequence spaces, Morrey sequence spaces, Jackson-type inequality}

\begin{abstract}
	In this paper we introduce positive order differences for sequences and study approximations in $l_p$ and
	Morrey sequence spaces. 
\end{abstract}

\maketitle

\section{Introduction}
Decomposition methods play an important role in the study of various function spaces. These methods allow us to obtain descriptions of such spaces that 
lead to the corresponding discrete spaces (see e.g.~\cite{N}, \cite{S}, \cite{T}). 
On the other hand, the study of analogues of classical operators in discrete harmonic analysis (see e.g. \cite{GS}, \cite{MSW}), is also an incentive to 
study discrete spaces.
So the consideration of discrete spaces is quite natural and is of independent interest.
In this paper we consider approximations in $l_p$ and in more general Morrey sequence spaces.

\section{Approximations in   $l_p$ spaces}\label{S:lp}

By $l_p$, $0<p\le \infty$, we denote the vector space of complex--valued sequences ${\bf a}=(a_1,a_2,\ldots)=(a_k)$	with finite (quasi)norm			 
         $$
\|{\bf a}\|_p=
\begin{cases}
\left(\sum_{n=1}^{\infty}|a_n|^p\right)^{1/p},  &\text{if $0<p<\infty$;}\\
\sup\limits_{n\ge 1}|a_n|,                                     &\text{if $p=\infty$}.
\end{cases}		 
		 $$
		 
The subspace of $l_\infty$ of all sequences which tend to zero with the same norm is denoted by $c_0$.

%By ${\bf e}_n$, $n=1,2,\ldots$, are denoted the vectors of the standart basis in $l_p$, $0<p<\infty$, and $c_0$ i.e. ${\bf e}_n=(\delta_{kn})$,
%where $\delta_{kn}$ is the Kronecker delta.
Let ${\bf e}_n=(\delta_{kn})$ where $\delta_{kn}$ is the Kronecker delta (these vectors form a basis both in $l_p$, $0<p<\infty$, and $c_0$) and 
$X_m={\rm span}({\bf e}_1,\ldots,{\bf e}_m)$, $m=1,2,\ldots$. For ${\bf a}\in l_p$ or ${\bf a}\in c_0$ the values
    $$
E_m({\bf a})_p=\inf\limits_{{\bf b}\in X_m}\|{\bf a}-{\bf b}\|_p
    $$
are called the best approximations of the vector ${\bf a}$ by elements from the linear spaces $X_m$.

We will denote by $\#A$ (or $|A|$) the number of elements of a finite set A.

Next, we need a notion similar to the notion of the difference of a function with a given step $h$.	
The duality between the function space $L_2(-\pi,\pi)$ and the sequence space $l_2$ under the Fourier transform, and
the relation between the modulus of smoothness of a given function and the norm of the trigonometric series with  coefficients 
$\min(1,(kh)^r)c_k$ (where $(c_k)$ are the Fourier coefficients of the original function) as obtained in \cite{Ci},
lead us to the following natural definition.

\begin{dfn}\label{D:diff}
 The difference $\Delta^r_h{\bf d}$ of the positive order $r>0$ with the step $h\ge 0$ for a sequence ${\bf d}$ is called
$\Delta^r_h{\bf d}=(\min(1,(kh)^r)d_k)$.
\end{dfn}

%See also   the relation between the moduli of smoothness of a given function and the norm of the trigonometric series with coefficients 
%$(\min(1,(kh)^r)c_k)$ ($(c_k)$ is the sequence of Fourier coefficients of the original function) in [2].
%a formula expressing the moduli of smoothness of a function through the sequence $(\min(1,(kh)^r)c_k)$ ($(c_k)$ denotes 
%$the sequence of Fourier coefficients of a given function) was obtained.

Recall the definitions of the derivative of a sequence and Sobolev sequence spaces. The $r$--th derivative ($r>0$) of the sequence ${\bf d}=(d_k)$ is defined as
${\bf d}^{(r)}=(k^rd_k)$.  The Sobolev sequence space of the order $r$ is denoted by $W^r(l_p)$ and consists of all sequences ${\bf a}$ such that ${\bf a}^{(r)}\in l_p$.
The expression $\|{\bf a}\|_{W^r(l_p)}=\|(k^r|a_k|)\|_p$ is a (quasi)norm on the $W^r(l_p)$.

\begin{prop}\label{P:diff} 
For any sequence ${\bf a}\in l_p$ its differences have the following properties

   \begin{itemize}

\item[1)] $\|\Delta^r_h{\bf a}\|_p\le\|{\bf a}\|_p$, if $0<h<1$, and $\|\Delta^r_h{\bf a}\|_p=\|{\bf a}\|_p$, if $h\ge 1$;

\item[2)] $\lim_{h\to 0}\|\Delta^r_h{\bf a}\|_p=0$ if $p<\infty$; $\lim_{h\to 0}\|\Delta^r_h{\bf a}\|_\infty=0$ if ${\bf a}\in c_0$;

\item[3)] if $0<h_1<h_2$ and $0<p\le\infty$, then $\|\Delta^r_{h_1}{\bf a}\|_p\le \|\Delta^r_{h_2}{\bf a}\|_p$;

\item[4)] if $\lambda>1$ and $0<p\le\infty$, then $\|\Delta^r_{\lambda h}{\bf a}\|_p\le ([\lambda]+1)^r\|\Delta^r_h{\bf a}\|_p$, where $[\lambda]$ 
denotes the integer part of the number $\lambda$;

\item[5)] if ${\bf a}\in W^s(l_p)$, $0<p\le\infty$  and  $r>s$, then  $\|\Delta^r_h{\bf a}\|_p\le h^s\|\Delta^{r-s}_h\bf{a^{(s)}}\|_p$ and 
$\|\Delta^r_h{\bf a}\|_p\le h^r\|{\bf a}^{(r)}\|_p$;

\item[6)] if $r>s$ and $0<p\le\infty$, then $\|\Delta^r_h{\bf a}\|_p\le \|\Delta^s_h{\bf a}\|_p$.

  \end{itemize}
\end{prop}  

\begin{proof} 
1) Immediately follows from the definition.

2) Given $\varepsilon>0$ there exists $m_\varepsilon\in\mathbb{N}$ such that $\sum_{k=m_\varepsilon+1}^\infty |a_k|^p<\varepsilon^p$ and $1/m_\varepsilon^{r}<\varepsilon$.
Let $h<1/m_\varepsilon^2$, then for $k\in\{1,\dots,m_\varepsilon\}$ we have 
$\min(1,(kh)^r)<1/m_\varepsilon^r$. Hence
$$
\|\Delta^r_h{\bf a}\|_p^p=\sum_{k=1}^{m_\varepsilon} (\min(1,(kh)^r)|a_k|)^p+\sum_{k=m_\varepsilon+1}^\infty (\min(1,(kh)^r)|a_k|)^p<
$$
   $$
\frac{1}{m_\varepsilon^{rp}}\sum_{k=1}^{m_\varepsilon}|a_k|^p+\sum_{k=m_\varepsilon+1}^\infty |a_k|^p<\varepsilon^p \|{\bf a}\|^p_p+\varepsilon^p,
   $$
which proves 2) for $0<p<\infty$.

For sequences from $c_0$, the proof requires an obvious modification.

3) For $0<h_1<h_2$ we have $\min(1,(kh_1)^r)\le \min(1,(kh_2)^r)$ from which follows the validity of the estimate.

4) Let us first consider the case when $\lambda=l\in\mathbb{N}$ and $l\ge 2$. Suppose that $lh\le 1$, then
$$
\|\Delta^r_{lh}{\bf a}\|_p^p=\sum_{k=1}^{[1/(lh)]} (\min(1,(klh)^r)|a_k|)^p+\sum_{[1/lh]+1}^{k=[1/h]} (\min(1,(klh)^r)|a_k|)^p+
$$
   $$
\sum_{k=[1/h]+1}^\infty (\min(1,(klh)^r)|a_k|)^p=S_1+S_2+S_3.
   $$
For $k\in\{1,\ldots,[1/(lh)]\}$ we have
$\min(1,(klh)^r)=l^r\min(1,(kh)^r)$, and for $k\in\{[1/(lh)]+1,\ldots,[1/h]\}$ we have $1/l<kh\le 1$. Then
 $$
S_1=l^{rp}\sum_{k=1}^{[1/(lh)]} (\min(1,(kh)^r)|a_k|)^p,
 $$
    $$
S_2=\sum_{k=[1/(lh)]+1}^{[1/h]} |a_k|^p\le l^{rp}\sum_{k=[1/lh]+1}^{[1/h]}((kh)^r|a_k|)^p=
l^{rp}\sum_{k=[1/(lh)]+1}^{[1/h]}(\min(1,(kh)^r)|a_k|)^p,
    $$
$$
S_3\le l^{rp}\sum_{k=[1/h]+1}^\infty |a_k|^p.
$$ 
The obtained inequalities lead to the inequality $\|\Delta^r_{lh}{\bf a}\|_p^p\le l^{rp}\|\Delta^r_{h}{\bf a}\|_p^p$. 

Suppose that $lh\ge 1$. If $h\ge 1$, then the estimate is obvious. For $0<h<1$ we have
$$
\|\Delta^r_{lh}{\bf a}\|_p^p=\|{\bf a}\|^p_p=\sum_{k=1}^{[1/h]}|a_k|^p+\sum_{k=[1/h]+1}^\infty |a_k|^p\le
$$
   $$
\sum_{k=1}^{[1/h]}(klh)^{rp}|a_k|^p+\sum_{k=[1/h]+1}^\infty l^{rp}|a_k|^p=l^{rp}\|\Delta^r_h{\bf a}\|_p^p.
   $$   
  
Now let $\lambda>1$ be arbitrary. Since $1\le [\lambda]\le \lambda<[\lambda]+1$, then using property 3) and the inequality obtained above  
we have
$$
\|\Delta^r_{\lambda h}{\bf a}\|_p\le \|\Delta^r_{([\lambda]+1)h}{\bf a}\|_p\le
([\lambda h]+1)^{r}\|\Delta^r_h{\bf a}\|_p.
$$

5) It is obvious that the estimates are valid for $h\ge 1$. Let $0<h<1$ and $k\in\{1,\ldots,[1/h]\}$. Then 
$\min(1,(kh)^r)=(kh)^{s}\min(1,(kh)^{r-s})$. Therefore
$$
\|\Delta^r_h{\bf a}\|_p^p\le h^{sp}\sum_{k=1}^{[1/h]} (\min(1,(kh)^{r-s})k^s|a_k|)^p+ 
h^{sp}\sum_{k=[1/h]+1}^\infty (k^s|a_k|)^p=h^{sp}\|\Delta^{r-s}_h\bf{a^{(s)}}\|_p^p.
$$

The second inequality follows directly from the definitions.

6) The inequality is a consequence of $\min(1,(kh)^r)\le \min(1,(kh)^s)$.
\end{proof}

The basic inequalities of approximation theory, such as Jackson's and Bernstein's inequalities (see e.g.~\cite{DeV}, \cite{N}), remain valid 
for approximations by the elements from the spaces $X_m$. In addition, there are discrete analogues of other important inequalities.

\begin{prop}\label{P:ineq} 
The following inequalities are valid
        \begin{itemize}
\item[1)] for all $n\in\mathbb{N}$, $r>0$ and $0<p\le \infty$
$$
 E_n({\bf a})_p\le \|\Delta^r_{1/n}{\bf a}\|_p \mbox{ (Jackson's inequality)};
$$

\item[2)] for all $r>0$, $0<p\le \infty$ and ${\bf a}$ such that ${\rm supp}{\bf a}=\{k\in\mathbb{N}|\; a_k\ne 0\}\subset \{1,2,\ldots,N\}$
$$
\|{\bf a}^{(r)}\|_p\le N^r\|{\bf a}\|_p \mbox{ (Bernstein's inequality)};
$$

\item[3)] for all $r>0$, $0<p\le \infty$ and ${\bf a}\in l_p$ such that ${\rm supp}{\bf a}\subset \{N,N+1,\ldots\}$
$$
\|{\bf a}\|_p\le N^{-r}\|{\bf a}^{(r)}\|_p \mbox{ (B\"{o}hr--Favard's inequality)};
$$

\item[4)] for all $r>0$ and ${\bf a}$ such that ${\rm supp}{\bf a}\subset \{N,N+1,\ldots\,2N\}$
$$
(2N)^{-r}\|{\bf a}^{(r)}\|_p \le \|\Delta^r_{1/N}{\bf a}\|_p=\|{\bf a}\|_p\le N^{-r}\|{\bf a}^{(r)}\|_p;
$$

\item[5)] for all  ${\bf a}$ such that ${\rm supp}{\bf a}\subset \{1,2,\ldots,N\}$ and $0<p<q\le \infty$
$$
\|{\bf a}\|_p\le (\# {\rm supp}{\bf a})^{1/p-1/q}\|{\bf a}\|_q \mbox{ (Nikolskii's inequality)}.
$$
        \end{itemize}
\end{prop}		

\begin{proof} 
Since
$$
E_n({\bf a})_p=\left(\sum_{k=n+1}^{\infty}|a_k|^p\right)^{1/p},\; 0<p<\infty, \mbox{ and } 
E_n({\bf a})_\infty=\sup\limits_{k>n}|a_k|, 
$$
then  for both $0<p<\infty$ and $p=\infty$ we have $E_n({\bf a})_p\le \|\Delta^r_{1/n}{\bf a}\|_p$.

2) The evaluation is obvious.

3) Indeed, for positive $r$ we have
$\|(a_k)\|_p=\|(k^{-r}k^{r}a_k)\|_p\le N^{-r}\|(a_k)\|_p$.

4) The required estimates follow from 3) and 2).

5) By H\"{o}lder's inequality with the exponent $q/p$
$$
\left(\sum_{k\in{\rm supp}{\bf a}}|a_k|^p\right)^{1/p}\le (\# {\rm supp}{\bf a})^{1/p-1/q} \|{\bf a}\|_q.
$$
\end{proof}

\begin{remark} For ${\bf a}\in l_\infty$ the following statements are equivalent
    \begin{itemize}
\item[1)]  ${\bf a}\in c_0$;

\item[2)] $\lim_{h\to 0}\|\Delta^r_h{\bf a}\|_\infty=0$;

\item[3)] $E_n({\bf a})_\infty\to 0$ $(n\to \infty)$.
    \end{itemize}

The assertion follows from the implications 1) $\Rightarrow$ 2) $\Rightarrow$ 3) $\Rightarrow$ 1).
\end{remark}

The introduced differences allow us to give a description of approximation sequence spaces considered in~\cite{P} 
in the framework of abstract approximation schemes.

Let $r>0$, $0<p<\infty$ and $0<\theta\le \infty$. Then the approximation space $A^r_\theta(l_p)$ consists of all elements
${\bf a}\in l_p$ such that $(k^{r-1/\theta}E_k({\bf a})_p)\in l_\theta$. The (quasi)norm in $A^r_\theta(l_p)$ is defined as follows
$\|{\bf a}\|_{A^r_\theta(l_p)}=\|(k^{r-1/\theta}E_k({\bf a})_p)\|_\theta+\|{\bf a}\|_p$. These spaces can be considered as the discrete
counterpart of the Nikolskii--Besov function spaces. But it is known that Nikolskii--Besov spaces primarily were defined through differences of
higher orders \cite{N}. Therefore, $B^r_\theta(l_p)$, $r>0$, $0<p,\theta\le\infty$ denotes the Nikolskii--Besov sequence space of all $l_p$ sequences
for which the expression (which is a (quasi) norm on $B^r_\theta(l_p)$)
$$
\|{\bf a}\|_{\it{B}^r_\theta(l_p)}=\left(\sum_{k=1}^\infty[k^r\|\Delta^s_{1/k}{\bf a}\|_p]^\theta\frac{1}{k}\right)^{1/\theta}
$$
(when $\theta=\infty$ the sum is replaced by $\sup_{k\ge 1}(k^r\|\Delta^s_{1/k}{\bf a}\|_p)$)
is finite for some $s>r$.

\begin{thm}
 Spaces $A^r_\theta(l_p)$ and $B^r_\theta(l_p)$ coincide with the equivalence of (quasi)norms.
\end{thm} 

\begin{proof} 
The estimate $\|{\bf a}\|_{A^r_\theta(l_p)}\le \|{\bf a}\|_{B^r_\theta(l_p)}$ follows from Jackson's inequality.
The opposite  inequality can be obtained by using the technique from~\cite{N} and we will not dwell on its details.
\end{proof}

The following statement is a counterpart of the relation obtained in \cite{JS}. Let $t>0$ and
$$
K({\bf a},t;l_p,W^r(l_p))=\inf\limits_{{\bf d}\in W^r(l_p)}(\|{\bf a-d}\|_p+t^r\|{\bf d}\|_{W^r(l_p)})
$$
be a K--functional of the element ${\bf a}\in l_p$. Then 
$C_1 \|\Delta^r_t{\bf a}\|_p\le K({\bf a},t^r;l_p,W^r(l_p))\le C_2\|\Delta^r_t{\bf a}\|_p$ with constants independent of ${\bf a}$ and $t$. 
A direct proof of these estimates is effortless.
%Proof of these estimates is effortless.
%Here is another statement, which is a counterpart of Arzel's and Riesz's theorems (criterion of compactness in $L_p$ spaces).

The differences allow us to reformulate compactness criterion in the $l_p$ spaces. 

\begin{dfn} 
The sequences of a set $A\subset l_p$ are called equicontinuos if for some $r>0$ for every $\varepsilon>0$ there is $\delta>0$
such that for all $0<h<\delta$, the relation $\|\Delta_h^r{\bf a}\|_p<\varepsilon$ holds for every sequence ${\bf a}\in A$.
\end{dfn}

\begin{thm} Let $D\subset l_p$, $1\le p<\infty$, or $D\subset c_0$ be a closed set. Then $D$ is compact if and only if $D$ is bounded 
and the sequences ${\bf a}$ from the set $D$ are equicontinuos.
\end{thm}

\begin{proof}
This assertion follows from the criterion of compactness in $l_p$ \cite{LS} and the second statement of~\ref{P:diff}.
\end{proof}

\section{Approximations in Morrey sequence spaces}

The main properties of Morrey sequence spaces are given in~\cite{GK}, for Morrey function spaces see e.g.~\cite{SFH}.  
Here we consider approximations similar to those in~\ref{S:lp}, namely by sequences with supports in segments of integers symmetric about zero.

The symbols $\mathbb{N}$ and $\mathbb{Z}$ as usual denote the sets of natural and integer numbers, respectively.
Let  $\mathbb{N}_0=\mathbb{N}\cup\{0\}$, $m\in\mathbb{Z}$, $B(m,N)=\{m-N,m-N+1,\ldots,m+N\}$ be a ball centered at $m$ and radius $N$,
$|B(m,N)|=2N+1$. For any set $E\subset \mathbb{Z}$, by definition ${\bf a}\chi_E=(b_k)$, where $b_k=a_k$, if $k\in E$ and $b_k=0$, if $k\notin E$.
%Also put ${\bf a}\chi_{B(m,N)}=(b_k)$, where $b_k=a_k$, if $|k|\in B(m,N)$ and $b_k=0$, if $k\notin B(m,N)$.

The Morrey sequence space $m^{q,p}$, $0<p\le q<\infty$, is the collection of all complex valued sequences ${\bf a}=(a_k)$, $k\in\mathbb{Z}$, for which 
$$
\|{\bf a}\|_{m^{q,p}}=\sup_{m\in\mathbb{Z},\;N\in\mathbb{N}_0}|B(m,N)|^{1/q-1/p}\left(\sum_{k\in B(m,N)}|a_k|^p\right)^{1/p}=
$$
    $$
\sup_{m\in\mathbb{Z},\;N\in\mathbb{N}_0}(2N+1)^{1/q-1/p}\|{\bf a}\chi_{B(m,N)}\|_p<\infty.
    $$
These spaces are Banach (quasi) normed spaces \cite{GK}. Note that for p=q the spaces $m^{q,p}$ coincide with $l_q$. 


For sequences ${\bf a}=(a_k)_{k\in\mathbb{Z}}$, the differences of order $r>0$ with step $0\le h$ and the $r$--th derivative of order r are defined by
$\Delta^r_h{\bf a}=(\min(1,|\tilde{k}h|^r)a_k)_{k\in\mathbb{Z}}$, ${\bf a}^{(r)}=(\tilde{|k|}^ra_k)_{k\in\mathbb{Z}}$, where
$\tilde{k}=\tilde{k}(k)=1$, if $k=0$, and $\tilde{k}=k$, if $k\ne 0$.

In the following, the sums over empty sets are assumed to be zero.


Let ${\bf a}\in m^{q,p}$. Then for any ${\bf b}$ with ${\rm supp}{\bf b}\subset B(0,L)$  
$$
\|({\bf a}-{\bf b})\chi_{B(m,N)}\|_p=\left(\sum_{k\in B(m,N),\:|k|\le L}|a_k-b_k|^p+\sum_{k\in B(m,N),\:|k|>L}|a_k|^p\right)^{1/p}\ge
$$
   $$
\left(\sum_{k\in B(m,N),\:|k|>L}|a_k|^p\right)^{1/p}=\|{\bf a}\chi_{B(m,N)\cap (\mathbb{Z}\setminus B(0,L))}\|_p.
   $$
Hence, 
$$
E_L({\bf a})_{m^{q,p}}=\inf_{{\bf b},\:{\rm supp}{\bf b}\subset B(0,L)}\|{\bf a}-{\bf b}\|_{m^{q,p}}=\|{\bf a}\chi_{\mathbb{Z}\setminus{B(0,L)}}\|_{m^{q,p}}.
$$ 

Below, for any $r>0$
$$
\left(\sum_{k\in B(m,N),\:|k|>L}|a_k|^p\right)^{1/p}=\left(\sum_{k\in B(m,N),\:|k|>L}(\min(1,|k/L|^r)|a_k|)^p\right)^{1/p}\le
$$
   $$
\|\Delta^r_{1/L}{\bf a}\chi_{B(m,N)}\|_p                    %\left(\sum_{k\in B(m,N)}(\min(1,|k/L|^r|a_k|)^p\right)^{1/p}=.
   $$
From this follows  Jackson's inequality 
$$
E_L({\bf a})_{m^{q,p}}\le \|\Delta^r_{1/L}{\bf a}\|_{m^{q,p}}.
$$

It is known [3] that ${\bf d}=(1/|k|^\alpha)_{k\in\mathbb{Z}}\in m^{q,p}$ if $q=1/\alpha$, $0<p<q<\infty$ (for certainty we assume that $d_0=1$). Let us show that
$\|{\bf d}-{\bf d}\chi_{B(0,k)}\|_{m^{q,p}}\nrightarrow 0$ ($k\to\infty$).

Indeed, 
$$
\|{\bf d}-{\bf d}\chi_{B(0,2^j)}\|_{m^{q,p}}\ge (2^{j+2}+1)^{1/q-1/p}\|{\bf d}\chi_{\mathbb{Z}\setminus B(0,2^j)}\chi_{B(0,2^{j+1})}\|_p\ge
$$
   $$
2^{(j+2)(1/q-1/p)}\left(\sum_{|k|=2^j+1}^{2^{j+1}}\frac{1}{|k|^{\alpha p}}\right)^{1/p}\ge 2^{1+\alpha-2/p}.
   $$

\begin{thm}
 $E_L({\bf a})_{m^{q,p}}\to 0$ ($L\to\infty$) if and only if $\|\Delta^r_h{\bf a}\|_{m^{q,p}}\to 0$ ($h\to 0$).
\end{thm}

\begin{proof}
Sufficiency follows from Jackson's inequality. Let us proceed to the proof of necessity. By condition
$E_L({\bf a})_{m^{q,p}}\to 0$ ($L\to\infty$), i.e. $\|{\bf a}\chi_{\mathbb{Z}\setminus B(0,L)}\|_{m^{q,p}}\to 0$. Hence,
for a given $\varepsilon>0$ there exists $L_\varepsilon\in\mathbb{N}$ such that 
$\|{\bf a}\chi_{\mathbb{Z}\setminus B(0,L_\varepsilon-1)}\|_{m^{q,p}}<\varepsilon$ and $1/L_\varepsilon^r<\varepsilon$. Let $|h|<1/L_\varepsilon^2$.
For
$$ 
I=\sum_{k\in B(m,N)}(\min(1,|\tilde {k}h|^r)|a_k|)^p=
$$
   $$
\sum_{k\in B(m,N),\: |k|< L_\varepsilon}(\min(1,|\tilde {k}h|^r)|a_k|)^p +
\sum_{k\in B(m,N)\: |k|\ge L_\varepsilon}(\min(1,|\tilde {k}h|^r)|a_k|)^p
   $$
by virtue of $|h|<1/L^2_\varepsilon$ we have
$$
I\le \frac{1}{L_\varepsilon^{rp}}\sum_{k\in B(m,N),\: |k|< L_\varepsilon}|a_k|^p+\sum_{k\in B(m,N),\: |k|\ge L_\varepsilon}|a_k|^p\le
$$
    $$
\varepsilon^p \sum_{k\in B(m,N)}|a_k|^p	+ \sum_{k\in B(m,N),\: |k|\ge L_\varepsilon}|a_k|^p.
	$$
Thus, for all $|h|<1/L^2_\varepsilon$ we have $\|\Delta^r_h{\bf a}\|_{m^{q,p}}\le C(\varepsilon \|{\bf a}\|_{m^{q,p}}+ \varepsilon)$.
\end{proof}


Thus, from the point of view of the approximations under consideration, the subspaces of $m^{q,p}$ for whose elements the differences 
tend to zero are of interest. Such subspaces for the Morrey function spaces were introduced in \cite{Z}.

\begin{prop}\label{P:Morreyineq}
The following inequalities hold
       \begin{itemize}
\item[1)] if ${\rm supp}{\bf b}\subset B(0,n_0)$, $n_0\in\mathbb{N}$, then for all $r>0$
$$
\|{\bf b}^{(r)}\|_{m^{q,p}}\le n_0^r\|{\bf b}\|_{m^{q,p}} \mbox{ (Bernstein's inequality)};
$$

\item[2)] if ${\rm supp}{\bf b}\subset\mathbb{Z}\setminus B(0,n_0-1)$, $n_0\in\mathbb{N}$, then for all $r>0$
$$
\|{\bf b}\|_{m^{q,p}}\le n_0^{-r}\|{\bf b}^{(r)}\|_{m^{q,p}} \mbox{ (B\"{o}hr--Favard's inequality)};
$$

\item[3)] if ${\rm supp}{\bf b}\subset B(0,2n_0)\setminus B(0,n_0-1)$, $n_0\in\mathbb{N}$, then for all $r>0$ 
$$
(2n_0)^{-r}\|{\bf b}^{(r)}\|_{m^{q,p}} \le \|\Delta^r_{1/n_0}{\bf b}\|_{m^{q,p}}=\|{\bf b}\|_p\le n_0^{-r}\|{\bf b}^{(r)}\|_{m^{q,p}};
$$

\item[4)] if ${\rm supp}{\bf b}\subset B(0,n_0)$, $n_0\in\mathbb{N}$ and $0<p_1<p_2\le q<\infty$ then
$$
\|{\bf b}\|_{m^{q,p_1}}\le (\# {\rm supp}{\bf b})^{1/p_1-1/p_2}\|{\bf b}\|_{m^{q,p_2}} \mbox{ (Nikolskii's inequality)}).
$$
      \end{itemize}
\end{prop}

\begin{proof} 
1)
From $\|{\bf b}^{(r)}\chi_{B(m,N)}\|_p\le n_0^r\|{\bf b}\chi_{B(m,N)}\|_p$ follows the validity of the statement.

%Let ${\rm supp}{\bf b}\subset \{-n_0,\ldots,n_0\}$. Then 
%$$
%\sum_{k\in B(m,N)}|k^rb_k|^p=\sum_{k\in B(m,N)\cap B(0,n_0)}|k^rb_k|^p\le
%$$
%    $$
%\max_{l\in B(0,n_0)}|l|^{rp}\sum_{k\in B(m,N)\cap B(0,n_0)}|b_k|^p=
%n_0^{rp}\sum_{k\in B(m,N)}|k^rb_k|^p
%    $$
%This yields Bernstein's inequality $\|{\bf b}^{(r)}\|_{m^{q,p}}\le n_0^r\|{\bf b}\|_{m^{q,p}}$.

2)
For positive $r$
$$
\|{\bf b}\chi_{B(m,N)}\|^p_p=\sum_{k\in B(m,N),\: |k|\ge n_0}|b_k|^p\le
$$
    $$
n_0^{-rp}\sum_{k\in B(m,N),\: |k|\ge n_0}|k^rb_k|^p
\le n_0^{-rp}\|{\bf b}^{(r)}\chi_{B(m,N)}\|^p_p
    $$
from which the inequality to be proved follows.

3)
This inequality follows from 1), 2) and~\ref{D:diff}.

4)
By  H\"{o}lder's inequality with the exponent $p_2/p_1$
$$
\|{\bf b}\chi_{B(m,N)}\|_{p_1}\le (\# {\rm supp}{\bf b})^{1/p_1-1/p_2}\|{\bf b}\chi_{B(m,N)}\|_{p_2}.
$$			
Then the required inequality follows from the obvious estimate 
$$
(2N+1)^{1/q-1/p_1}\le(2N+1)^{1/q-1/p_2}.
$$
\end{proof}

The results of this note were partly announced in \cite{K}.

\begin{thebibliography}{99}

\bibitem{DeV} R.A. DeVore and G.G. Lorentz, {\it Constructive approximation}, Springer-Verlag, Berlin Heidelberg, 1993.

\bibitem{Ci} Z. Ciesielski, {\it Bases and $K$--functionals for Sobolev spaces over compact manifolds of class $C^{\infty}$},
Orthogonal series and approximation of functions. Collection of articles. Dedicated to Academician N. N. Luzin on the occasion 
of the 100th anniversary of his birth. 
Trudy Mat. Inst. Steklov., \textbf{164} (1983), 197–202 (in Russian); Proc. Steklov Inst. Math., \textbf{164} (1985), 225–231

\bibitem{GK} H. Gunawan, E. Kikianty, C. Schwanke, {\it Discrete Morrey spaces and their inclusion properties}, Math.
Nachr. \textbf(291)(8–9) (2018), 1283–1296, 10.1002/mana.201500425.

\bibitem{GS} H. Gunawan, C. Schwanke, {\it The Hardy–-Littlewood Maximal Operator on Discrete Morrey Spaces}, Mediterr. J. Math. \textbf{16}, 24 (2019), 
10.1007/s00009-018-1277-7.

\bibitem{JS} H. Johnen, K. Scherer, {\it On the equivalence of the K--functional and moduli of 
continuity and some applications}, in: Constructive Theory of Functions of Several 
Variables. Lecture Notes in Mathematics, \textbf{571},  Springer, Berlin New York, 1976,  119-140.
 
\bibitem{K} N.L. Kudryavtsev, {\it On elements of approximation theory based on $l_p$ spaces}, in:  Function Spaces.
Differential Operators. Problems of Mathematical Edu\-cation, Abstracts: The 6th International Conference, dedicated to the centennial anniversary
of the corresponding member of Russian Academy of Sciences, academician of European Academy of Sciences L.D. Kudryavtsev, Moscow, Russia, November 14-19, 2023, 
RUDN University Moscow (2023), 57 (in Russian).

\bibitem{LS} L.A. Lusternik and V.I. Sobolev, {\it Elements of functional analysis}, 3rd English transl., Hindustan publishing corpor., Delhi, 1974.

\bibitem{MSW} A. Magyar, E. M. Stein, S. Wainger, {\it Discrete Analogues in Harmonic Analysis: Spherical Averages},
Annals of Mathematics, 2nd Series, \textbf{155}, No. 1 (2002), 189-208.

\bibitem{N} S.M Nikolskii, {\it Approximation of functions of several variables and imbedding theorems}, Springer Berlin Heidelberg, 2011.

\bibitem{P} A. Pietsch, {\it Approximation Spaces}, J. Approx. Theory \textbf{32} (1981), 115-134.

\bibitem{S} Y. Sawano, {\it Theory of Besov Spaces}, Springer Singapore, 2018.

\bibitem{SFH} Y. Sawano, G. Di Fazio and D.I. Hakim, {\it Morrey spaces: introduction and applications to integral operators and PDE's}, Volume I, 
Chapman and Hall/CRC, New York, 2020.

\bibitem{T} H. Triebel {\it Theory of Function Spaces III}, Birkhäuser Basel, 2006.

\bibitem{Z} C. Zorko, {\it Morrey spaces}, Proc. Amer. Math. Soc. \textbf{98} (1986), 586–592.

\end{thebibliography}
			
			\end{document}