The Approximation Characteristics of Weighted p-Wiener Algebra

Abstract

In this paper, we study the approximation characteristics of weighted p-Wiener algebra ${\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)$ for $1\le p<\infty$ defined on the d-dimensional torus ${\mathbb{T}}^d$. In particular, we investigate the asymptotic behavior of the approximation numbers, Kolmogorov numbers, and entropy numbers associated with the embeddings $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)$ and $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)$ for $1\le p$, $q<\infty$, where $\mathcal{A}\left({\mathbb{T}}^d\right)$ is the Wiener algebra defined on the d-dimensional torus ${\mathbb{T}}^d$.

1. Introduction

The approximation numbers, Kolmogorov numbers, entropy numbers, and associated widths serve as fundamental components in the examination of stability, conditioning, approximation, and sensitivity properties pertaining to linear operators and matrices. These concepts have proven indispensable across various fields encompassing mathematics, numerical analysis, and scientific computing.

Within the context of Pitsch’s framework [1], the approximation numbers and Kolmogorov numbers were considered as the s-numbers and there were certain rules assigning to every operator a decreasing sequence of numbers which characterized their approximation or compactness properties in Banach space. Extensive research has been devoted to investigating the s-numbers and their associated widths. The approximation number, a measure of how effectively a linear operator or matrix can be approximated by a low-rank matrix, finds application in diverse areas such as data compression, signal processing, and image reconstruction. Achieving a smaller approximation numbers corresponds to a higher quality approximation. The Kolmogorov numbers characterize the rate at which singular values of a linear operator or matrix decay. These prove valuable in the analysis of stability and conditioning of numerical algorithms, particularly in tasks such as solving linear systems or computing eigenvalues.

In addition to the conventional s-numbers, entropy numbers, also known as pseudo s-numbers, offer a more refined approach to analyzing the sensitivity of operators. Entropy numbers are mathematical quantities employed to assess the complexity of function classes or sets by means of covering numbers, which are the estimate of the approximation achieved by a given set K through finite elements [2]. The theory of entropy numbers, as summarized by Pietsch [3], highlights their versatility as a tool for analyzing the complexity, approximation, and generalization properties of function classes, sets, and statistical models. Entropy numbers have various applications in functional analysis, approximation theory, statistical learning theory, and machine learning.

The width theory constitutes a central topic within function approximation theory, revolving around the pursuit of the most optimal approximation set and method in a certain sense, with a primary focus on quantifying the approximation error of sets in a normed space. Kolmogorov [4] originally introduced the concept of Kolmogorov width, which serves to characterize the best approximation error achievable by the “worst” element within a function class. Subsequently, Tikhomirov [5] extended this framework by introducing the notion of linear width, contributing to the comprehension of the trade-off between accuracy and complexity in various computational tasks. Additionally, the linear width offers valuable insights into the behavior of linear operators under diverse conditions. It is worth noting that the Kolmogorov widths, the linear widths, and the pseudo-widths correspond respectively to the Kolmogorov numbers, the approximation numbers, and the entropy numbers of the corresponding identity operators. These measures play a vital role in assessing the approximation properties of linear operators, enabling researchers to analyze and understand the behavior of various function classes and operators in the context of function approximation. For a comprehensive understanding of s-numbers and their associated widths, please refer to the following References [1,3,6,7,8,9], and for the details about entropy numbers, see References [3,10,11,12,13,14,15].

Owing to the exceptional characteristics exhibited by s-numbers and pseudo s-numbers, an extensive body of scholarly work has been dedicated to their comprehensive investigation, focusing particularly on estimating the asymptotic behavior of these quantities in diverse function spaces. Zhang et al. [16] conducted a study examining the asymptotic behavior of Gelfand, Kolmogorov, and Weyl numbers within Sobolev embeddings in weighted function spaces, specifically those of the Besov and Triebel–Lizorkin types, incorporating polynomial weights.

Wiener algebra refers to the set of all integrable functions with absolutely convergent Fourier series, which forms a Banach space equipped with a norm ${\parallel f\parallel }_{\mathcal{A}\left({\mathbb{T}}^d\right)}=:{\displaystyle \sum _{k\in {\mathbb{Z}}^d}}|\widehat{f}\left(k\right)|$. Cobos et al. [17] studied approximation numbers of embeddings of Sobolev spaces with mixed smoothness in the norm of ${\parallel ·\parallel }_{\mathcal{A}\left({\mathbb{T}}^d\right)}$. Byrenheid et al. [18] studied sample widths of multivariate periodic function classes in the norm of ${\parallel ·\parallel }_{\mathcal{A}\left({\mathbb{T}}^d\right)}$. Sickel et al. [19] established the accurate asymptotic behavior of the s-numbers associated with embeddings in weighted Wiener algebraic spaces. Their work facilitated a reduction of the approximation numbers and Kolmogorov numbers for bounded linear operators to the corresponding numbers for diagonal operators between sequences. In other words, they provided a profound insight into the relationship between the approximation characteristics of linear operators acting on function spaces and those acting on sequences, specifically diagonal operators. This analysis yielded valuable results regarding the asymptotic orders of these numbers and their interconnections within the context of weighted Wiener algebraic spaces.

These investigations contribute to our understanding of the asymptotic properties of s-numbers and pseudo s-numbers and provide valuable insights into their behavior in various functional and function space contexts. However, the estimation of s-numbers and pseudo s-numbers within the framework of weighted p-Wiener algebra remains an unexplored subject that has not yet been investigated. This paper builds upon prior research and delves into an investigation of the approximation properties of weighted p-Wiener algebras. Initially, we propose the concept of the weighted p-Wiener algebra, drawing upon the foundation laid by the classical Wiener algebra. Subsequently, we examine the asymptotic behavior of the approximation numbers, Kolmogorov numbers, and entropy numbers associated with the embeddings $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)$ and $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)$, where $1\le p,q<\infty$. Additionally, we take inspiration from the work of Sickel et al. [19], which explored the asymptotic behavior of s-numbers related to the embeddings $id:{\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)$ and $id:{\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)$ but only considered the cases where $p=1$ and $q=2$.

The present paper is structured as follows. Section 2 introduces some preliminaries that will be employed throughout this study. Section 3 proceeds to present an exposition on the the approximation numbers, Kolmogorov numbers, and entropy numbers of embeddings of weighted p-Wiener algebra. Finally, Section 4 provides a concise summary of the findings and conclusions derived from this research endeavor.

2. Preliminaries

Let ${\mathbb{T}}^d$ be the d-dimensional torus. It shows that the volume of ${\mathbb{T}}^d$ is 1 when it is equipped with the probability measure ${\left(2\pi \right)}^{-d}dx$, and several related embeddings of operator norms equal to 1. Let $L_p\left({\mathbb{T}}^d\right)$ be a classical p-th integrable Lebesgue space with the norm

$${\parallel f\parallel }_{L_p\left({\mathbb{T}}^d\right)}:=\left\{\begin{array}{cc}{\left(2\pi \right)}^{-d}{\left({\int }_{{\mathbb{T}}^d}{\left|f\left(x\right)\right|}^{p}dx\right)}^{1/p},\hfill & 1\le p<\infty ,\hfill \\ esssup\left|f\right|,\hfill & p=\infty .\hfill \end{array}\right.$$

and let ${\ell }_p\left({\mathbb{Z}}^d\right),1\le p\le \infty ,$ be a classical sequence space on ${\mathbb{Z}}^d$ with the usual norm

$${\parallel x\parallel }_{{\ell }_p\left({\mathbb{Z}}^d\right)}:=\left\{\begin{array}{cc}{\left({\sum }_{i=1}^{\infty }{\left|x_i\right|}^p\right)}^{1/p},\hfill & 1\le p<\infty ,\hfill \\ {\displaystyle {sup}_{1<i<\infty }}\left|x_i\right|,\hfill & p=\infty .\hfill \end{array}\right.$$

For every function $f\in L_1\left({\mathbb{T}}^d\right)$, we define the Fourier transform as

$$\widehat{f}\left(x\right):={\left(2\pi \right)}^{-d}{\int }_{{\mathbb{T}}^d}f\left(t\right)e^{itx}dt,$$

and $\widehat{f}\left(k\right),k\in {\mathbb{Z}}^d$ denote the Fourier coefficients of f.

Note that $L_2\left({\mathbb{T}}^d\right)$ is a Hilbert space with the orthonormal basis $\left\{e^{ikx}:k\in {\mathbb{Z}}^d\right\}$. Therefore, it holds that, for every $f\in L_2\left({\mathbb{T}}^d\right)$,

$${∥f∥}_{L_2\left({\mathbb{T}}^d\right)}^2={\left(2\pi \right)}^{-d}{{\int }_{{\mathbb{T}}^d}\left|f\right(x\left)\right|}^{2}dx={{\displaystyle \sum _{k\in {\mathbb{Z}}^d}}\left|\widehat{f}\left(k\right)\right|}^2.$$

Let $\mathcal{A}\left({\mathbb{T}}^d\right)$ denote the classical Wiener algebra defined as

$$\mathcal{A}\left({\mathbb{T}}^d\right):=\left\{f\in L_1\left({\mathbb{T}}^d\right):\left\{\widehat{f}\left(k\right)\right\}\in {\ell }_1\left({\mathbb{Z}}^d\right)\right\}.$$

Consider a sequence $\omega =\left\{{\omega }_k\right\}$, where ${\omega }_k>0$ for all $k\in {\mathbb{Z}}^d$. In this context, such a sequence is denoted as a “weight”. Additionally, let $1/\omega =\{1/{\omega }_k\}\in {\ell }_{\frac{p}{p-1}}\left({\mathbb{Z}}^d\right)$, for $1<p<\infty$; and $1/\omega =\{1/{\omega }_k\}\in {\ell }_{\infty }\left({\mathbb{Z}}^d\right)$, for $p=1$. Under these conditions, we formally introduce the notion of the weighted p-Wiener algebra ${\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)$ as

$${\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right):=\left\{f\in L_1\left({\mathbb{T}}^d\right):\left\{{\omega }_k\widehat{f}\left(k\right)\right\}\in {\ell }_p\left({\mathbb{Z}}^d\right)\right\},1\le p<\infty .$$

Note that ${\mathcal{A}}_{\omega }^1\left({\mathbb{T}}^d\right):={\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right).$

Remark 1.

For $1\le p<\infty$, it is easy to prove that weighted p-Wiener algebra ${\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)$ is a Banach space equipped with the norm ${\parallel f\parallel }_{{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)}:={∥\left\{{\omega }_k\widehat{f}\left(k\right)\right\}∥}_{{\ell }_p\left({\mathbb{Z}}^d\right)},$ and there exists the embedding of ${\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\hookrightarrow \mathcal{A}\left({\mathbb{T}}^d\right)$.

In fact, for every function f belonging to the weighted p-Wiener algebra ${\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)$, the following inequality holds

$$\parallel \left\{\widehat{f}\left(k\right)\right\}\parallel {}_{{\ell }_1\left({\mathbb{Z}}^d\right)}={∥\left\{\frac{{\omega }_k\widehat{f}\left(k\right)}{{\omega }_k}\right\}∥}_{{\ell }_1\left({\mathbb{Z}}^d\right)}\le {∥\left\{{\omega }_k\widehat{f}\left(k\right)\right\}∥}_{{\ell }_p\left({\mathbb{Z}}^d\right)}·{∥\frac{1}{\omega }∥}_{{\ell }_{\frac{p}{p-1}}\left({\mathbb{Z}}^d\right)},$$

which implies that the sequence $\left\{\widehat{f}\left(k\right)\right\}\in {\ell }_1\left({\mathbb{Z}}^d\right)$. Consequently, we can deduce that f also belongs to the space $\mathcal{A}\left({\mathbb{T}}^d\right)$. As a result, we establish the existence of the embedding of ${\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\hookrightarrow \mathcal{A}\left({\mathbb{T}}^d\right)$.

Now we recall definitions of the approximation numbers, Kolmogorov numbers, and entropy numbers of linear bounded operators as well as their associated properties, which will play a crucial role in our investigations.

Let X, Y be two Banach spaces and T be a linear bounded operators from X to Y and $n\in \mathbb{N}$. Then the n-th approximation number of T is defined as

$$a_n\left(T\right)=inf\{\parallel T-A\parallel :\mathrm{A}\in \mathcal{L}(X,Y),rankA<n\},$$

where $\mathcal{L}(X,Y)$ contains all linear bounded operators from X to Y.

The n-th Kolmogorov number of T is defined as

$$d_n\left(T\right)={\displaystyle \underset{L_{n-1}}{inf}}{\displaystyle \underset{\parallel x\parallel \le 1}{sup}}{\displaystyle \underset{y\in L_{n-1}}{inf}}{\parallel Tx-y\parallel }_Y,$$

where the left-most infimum is taken over all subspaces $L_{n-1}$ in Y with the dimensional at $mostn-1.$

The n-th entropy number of linear bounded operator T is defined as

$${\epsilon }_n\left(T\right)=inf\left\{r>0:T\left(B_X\right)\subseteq {\displaystyle \bigcup _{i=1}^{2^{n-1}}}{B}_Y\left(y_i;r\right),y_1,\dots ,y_{2^{n-1}}\in Y\right\},$$

where $B_X$ is the unit ball within the space X, while $B_Y\left(y_i;r\right)$ denotes the ball centered at $y_i$ with a radius of r within the space Y.

The n-th approximation number and the n-th Kolmogorov number of linear bounded operator T are consider as s-numbers and the n-th entropy number of linear bounded operator T is also named as pseudo s-number.

Convention: For the sake of brevity and clarity, we adopt the notation $s_n\left(T\right)$ in place of $a_n\left(T\right)$, $d_n\left(T\right)$, and ${\epsilon }_n\left(T\right)$ throughout this work. Therefore, when we refer to $s_n\left(T\right)$ possessing certain properties $x,y,z$, it implies that these properties are applicable to all three quantities, namely, the approximation number, the Kolmogorov number, and the entropy number associated with the linear operator T.

The entropy numbers and s-numbers display a number of advantageous characteristics, especially the ideal property in the following lemma, which will be used to prove the main results. Further details on these properties can be found in the works of Pietsh [1,3], Milaré [14], Mayer [15], Dinh Dung [20], Temlyakov [21], and Belinsky [22].

Lemma 1

([1,3,21]). Let X, Y, $X_0$, $Y_0$ be Banach spaces, R, S, T be linear operators satisfying T $\in \mathcal{L}\left(X_0,X\right)$, $S\in \mathcal{L}(X,Y)$, $R\in \mathcal{L}\left(Y,Y_0\right)$, and $m,n\in \mathbb{N}$. Then the approximation numbers, the Kolmogorov numbers, and the entropy numbers have the ideal property as

$$s_n\left(RST\right)\le \parallel R\parallel ·s_n\left(S\right)·\parallel T\parallel .$$

In this paper, it is assumed that the positive constants c, $c_1$, $c_2$ depend solely on the parameters $p,q$, and d. For two positive functions $a\left(y\right)$ and $b\left(y\right)$, $y\in D$, we employ the notations $a\left(y\right)\preccurlyeq b\left(y\right)$ to indicate the existence of a constant c such that $a\left(y\right)\le c·b\left(y\right)$ holds for all y in D, and $a\left(y\right)\asymp b\left(y\right)$ to indicate the existence of constants $c_1$ and $c_2$ such that $c_1·b\left(y\right)\le a\left(y\right)\le c_2·b\left(y\right)$ holds for all y in D.

3. The Estimation of the Approximation Numbers, Kolmogorov Numbers, and Entropy Numbers of Embeddings of Weighted p-Wiener Algebra

This section aims to explore two different cases of the approximation numbers, Kolmogorov numbers, and entropy numbers concerning weighted p-Wiener algebra, namely, $s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)$ and $s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)$, for $1\le p,q<\infty$.

3.1. The Approximation Numbers, Kolmogorov Numbers, and Entropy Numbers of $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)$, $1\le p<\infty$

As postulated earlier, the sequence $\left\{{\omega }_k\right\}$ is considered to be a weight sequence. Additionally, we introduce the condition ${lim}_{{\left|k\right|}_1\to \infty }\left|{\omega }_k\right|=\infty$ to ensure the compactness of certain embeddings of $L_q\left({\mathbb{T}}^d\right),1\le q<\infty$. Let ${\left\{{\sigma }_n\right\}}_{n\in \mathbb{N}}$ be the non-increasing rearrangement of the weight sequence ${\left\{1/{\omega }_k\right\}}_{k\in {\mathbb{Z}}^d}.$ It is evident that ${lim}_{n\to \infty }{\sigma }_n=0$. Moreover, the rearranged sequence ${\left\{{\sigma }_n\right\}}_{n\in \mathbb{N}}$ holds significant importance in the estimation of the approximation numbers, Kolmogorov numbers, and entropy numbers.

The following lemma represents the exact order of the approximation numbers, Kolmogorov numbers, and entropy numbers of certain diagonal operators in classical sequence spaces, which is crucial for the proof of the main theorems in this paper.

Lemma 2

([3,23]). Let $\sigma ={\left\{{\sigma }_i\right\}}_{i\in \mathbb{N}}$ be a diagonal operator satisfying the condition of $\left\{{\sigma }_i\right\}$ being a non-increasing sequence.

(1) If $1\le q<p\le \infty$ and $1/r=1/q-1/p$, then

$$a_n\left(\sigma :{\ell }_p\left(\mathbb{N}\right)\to {\ell }_q\left(\mathbb{N}\right)\right)=d_n\left(\sigma :{\ell }_p\left(\mathbb{N}\right)\to {\ell }_q\left(\mathbb{N}\right)\right)={\left({\displaystyle \sum _{i=n}^{\infty }}{\sigma }_i^r\right)}^{1/r}.$$

(2) If $1\le p=q\le \infty$, then

$$a_n\left(\sigma :{\ell }_p\left(\mathbb{N}\right)\to {\ell }_q\left(\mathbb{N}\right)\right)=d_n\left(\sigma :{\ell }_p\left(\mathbb{N}\right)\to {\ell }_q\left(\mathbb{N}\right)\right)={\sigma }_n.$$

(3) If $p=1$ and $q=2$, then

$$a_n\left(\sigma :{\ell }_1\left(\mathbb{N}\right)\to {\ell }_2\left(\mathbb{N}\right)\right)=d_n\left(\sigma :{\ell }_1\left(\mathbb{N}\right)\to {\ell }_2\left({\mathbb{Z}}^d\right)\right)={\displaystyle \underset{h\ge n}{sup}}{\left(\frac{h-n+1}{{\sum }_{i=1}^h{\sigma }_i^{-2}}\right)}^{1/2}.$$

(4) If $0<p,q\le \infty$, ${sup}_{i\ge n}\frac{{\sigma }_i}{{\sigma }_n}·{\left(\frac{i}{n}\right)}^{\alpha }<\infty ,$ for some $\alpha >max(1/q-1/p,0)$, then

$${\epsilon }_n\left(\sigma :{\ell }_p\left(\mathbb{N}\right)\to {\ell }_q\left(\mathbb{N}\right)\right)\asymp n^{1/q-1/p}{\sigma }_n.$$

Utilizing the ideal property of s-numbers and pseudo s-numbers, we are able to estimate the approximation numbers, Kolmogorov numbers, and entropy numbers of the operator $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)$, $1\le p<\infty$. This estimation can be translated into terms of the approximation numbers, Kolmogorov numbers, and entropy numbers associated with the operator $\sigma :{\ell }_p\left({\mathbb{Z}}^d\right)\to {\ell }_1\left({\mathbb{Z}}^d\right)$, $1\le p<\infty$, as elucidated in Lemma 2.

Theorem 1.

Let $\omega =\left\{{\omega }_k\right\}$ be a weight sequence and ${lim}_{{\left|k\right|}_1\to \infty }\left|{\omega }_k\right|=\infty$. Let the sequence ${\left\{{\sigma }_n\right\}}_{n\in \mathbb{N}}$ be the non-increasing rearrangement of the sequence ${\left\{1/{\omega }_k\right\}}_{k\in {\mathbb{Z}}^d}$.

(1) If $1<p<\infty$ and $1/r=1-1/p,$ then

$$a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)=d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)={\left({\displaystyle \sum _{i=n}^{\infty }}{\sigma }_i^r\right)}^{1/r},$$

$${\epsilon }_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)\asymp n^{1/r}{\sigma }_n.$$

(2) If $p=1$ and the sequence $\left\{{\sigma }_n\right\}$ satisfies the condition outlined in Lemma 2(4), then

$${\epsilon }_n\left(id:{\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)\asymp {\sigma }_n.$$

Proof. 

In order to investigate the approximation numbers, Kolmogorov numbers, and entropy numbers of the embeddings of weighted p-Wiener algebra for $1\le p<\infty$, we consider the following commutative diagram firstly

$$\begin{array}{cc}{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\stackrel{id}{⟶}& \mathcal{A}\left({\mathbb{T}}^d\right)\\ A\downarrow & \uparrow B\\ {\ell }_p\left({\mathbb{Z}}^d\right)\stackrel{D_{\omega }}{⟶}& {\ell }_1\left({\mathbb{Z}}^d\right)\end{array}$$

(1)

where the linear operators $A:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to {\ell }_p\left({\mathbb{Z}}^d\right),$ $D_{\omega }:{\ell }_p\left({\mathbb{Z}}^d\right)\to {\ell }_1\left({\mathbb{Z}}^d\right)$ and $B:{\ell }_1\left({\mathbb{Z}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)$ are defined as

$$\begin{array}{c}Af:={\left\{{\omega }_k\widehat{f}\left(k\right)\right\}}_{k\in {\mathbb{Z}}^d};\hfill \\ D_{\omega }\xi :={\left\{{\xi }_k/{\omega }_k\right\}}_{k\in {\mathbb{Z}}^d},\xi ={\left\{{\xi }_k\right\}}_{k\in {\mathbb{Z}}^d}\in {\ell }_p\left({\mathbb{Z}}^d\right);\hfill \\ B\xi \left(x\right):={\displaystyle {\sum }_{k\in {\mathbb{Z}}^d}}{\xi }_k{e}^{ikx},x\in {\mathbb{T}}^d.\hfill \end{array}$$

(2)

It is obvious that linear operators A and B are invertible, and the operator norm $\parallel A\parallel =\parallel B\parallel =1$. By employing the properties of the approximation numbers, Kolmogorov numbers, and entropy numbers outlined in Lemma 1, and considering the diagram (1) where $id=B{D}_{\omega }A$, we can establish an upper bound for the approximation numbers, Kolmogorov numbers, and entropy numbers associated with the embedding of weighted p-Wiener algebra as

$$\begin{array}{cc}\hfill s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)& \le \parallel A\parallel ·s_n\left(D_{\omega }:{\ell }_p\left({\mathbb{Z}}^d\right)\to {\ell }_1\left({\mathbb{Z}}^d\right)\right)·\parallel B\parallel \hfill \\ \hfill & =s_n\left(D_{\omega }:{\ell }_p\left({\mathbb{Z}}^d\right)\to {\ell }_1\left({\mathbb{Z}}^d\right)\right).\hfill \end{array}$$

(3)

On the other hand, we can modify the diagram as follows,

$$\begin{array}{cc}{\ell }_p\left({\mathbb{Z}}^d\right)\stackrel{D_{\omega }}{⟶}& {\ell }_1\left({\mathbb{Z}}^d\right)\\ A^{-1}\downarrow & \uparrow B^{-1}\\ {\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\stackrel{id}{⟶}& \mathcal{A}\left({\mathbb{T}}^d\right)\end{array}$$

(4)

Similarly, according to the ideal property of the approximation numbers, Kolmogorov numbers and entropy numbers in Lemma 1 and considering the diagram (4) where $id=B^{-1}{D}_{\omega }{A}^{-1}$ with $∥A^{-1}∥=∥B^{-1}∥=1$, we obtain the lower bound of the approximation numbers, Kolmogorov numbers, and entropy numbers of the embedding of weighted p-Wiener algebra as

$$\begin{array}{cc}\hfill s_n\left(D_{\omega }:{\ell }_p\left({\mathbb{Z}}^d\right)\to {\ell }_1\left({\mathbb{Z}}^d\right)\right)& \le ∥A^{-1}∥·s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)·∥B^{-1}∥\hfill \\ \hfill & =s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right).\hfill \end{array}$$

(5)

From inequalities (3) and (5) and ${\left\{{\sigma }_n\right\}}_{n\in \mathbb{N}}$ being the non-increasing rearrangement of ${(1/{\omega }_k)}_{k\in {\mathbb{Z}}^d}$, we have

$$\begin{array}{cc}\hfill s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)& =s_n\left(D_{\omega }:{\ell }_p\left({\mathbb{Z}}^d\right)\to {\ell }_1\left({\mathbb{Z}}^d\right)\right)\hfill \\ \hfill & =s_n\left(\sigma :{\ell }_p\left(\mathbb{N}\right)\to {\ell }_1\left(\mathbb{N}\right)\right).\hfill \end{array}$$

(6)

If $1<p<\infty$ and $1/r=1-1/p$, then according to Lemma 2(1), Lemma 2(4), and Equation (6), we obtain

$$a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)=d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)={\left({\displaystyle \sum _{i=n}^{\infty }}{\sigma }_i^r\right)}^{1/r},$$

and

$${\epsilon }_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)={\epsilon }_n\left(\sigma :{\ell }_p\left(\mathbb{N}\right)\to {\ell }_1\left(\mathbb{N}\right)\right)\asymp n^{1/r}{\sigma }_n.$$

If $p=1$, then according to Lemma 2(4) and Equation (6), we obtain

$${\epsilon }_n\left(id:{\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)={\epsilon }_n\left(\sigma :{\ell }_1\left(\mathbb{N}\right)\to {\ell }_1\left(\mathbb{N}\right)\right)\asymp {\sigma }_n.$$

Therefore, the conclusion of Theorem 1 is proved.  □

Remark 2.

Based on Sickel’s findings as presented in [19], it can be established that, in the case of $p=1$, the exact orders of the approximation numbers, Kolmogorov numbers, and entropy numbers of the embedding of weighted Wiener algebra corresponds precisely to ${\sigma }_n$.

Corollary 1.

Let $n\in \mathbb{N}$ and Λ be an indicator set in ${\mathbb{Z}}^d$ with the two properties:

(1) $\left|\mathsf{\Lambda }\right|=n-1$, where $\left|\mathsf{\Lambda }\right|$ is the cardinality of Λ.

(2) For $\forall k\in \mathsf{\Lambda }$ and $\forall \ell \notin \mathsf{\Lambda }$, then $\omega \left(k\right)\le \omega (\ell )$ holds.

Then, for every $f\in L_1\left({\mathbb{T}}^d\right)$, the operator $S_{\mathsf{\Lambda }}$ defined as $S_{\mathsf{\Lambda }}f\left(x\right):={\sum }_{k\in \mathsf{\Lambda }}\widehat{f}\left(k\right)e^{ikx},$ is the optimal operator of the approximation number $a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right),$ and

$$a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)={\displaystyle \underset{{\parallel f\parallel }_{{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)}\le 1}{sup}}{∥f-S_{\mathsf{\Lambda }}f∥}_{\mathcal{A}\left({\mathbb{T}}^d\right)}.$$

Proof. 

For every $f\in L_1\left({\mathbb{T}}^d\right)$, according to the conditions (1) and (2), we have

$$\begin{array}{cc}\hfill {∥f-S_{\mathsf{\Lambda }}f∥}_{\mathcal{A}\left({\mathbb{T}}^d\right)}& ={\displaystyle \sum _{k\in {\mathbb{Z}}^d\backslash \mathsf{\Lambda }}}\left|\widehat{f}\left(k\right)\right|\hfill \\ & \le {\displaystyle \underset{\ell \in {\mathbb{Z}}^d\backslash \mathsf{\Lambda }}{sup}}\frac{1}{\omega (\ell )}{\left|\right|f\left|\right|}_{{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)}={\sigma }_n{\left|\right|f\left|\right|}_{{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)}\hfill \\ & \le \left\{\begin{array}{cc}{\left({\displaystyle {\sum }_{i=n}^{\infty }}{\sigma }_i^r\right)}^{1/r}{\parallel f\parallel }_{{\mathcal{A}}_{\omega }^p\left(T^d\right)},\hfill & p>1,\hfill \\ {\sigma }_n{\left|\right|f\parallel }_{{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)},\hfill & p=1.\hfill \end{array}\right.\hfill \end{array}$$

Based on the estimation of the approximation number $a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)$ in Theorem 1, we obtain

$$\begin{array}{c}\hfill ∥id-S_{\mathsf{\Lambda }}∥\le \left\{\begin{array}{cc}{\left({\displaystyle {\sum }_{i=n}^{\infty }}{\sigma }_i^r\right)}^{1/r},\hfill & p>1,\hfill \\ {\sigma }_n,\hfill & p=1,\hfill \end{array}\right.\\ \hfill =a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right).\end{array}$$

(7)

By the definition of the approximation number $a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)$ and inequality (7), we have

$$\begin{array}{c}\hfill a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)={\displaystyle \underset{{\parallel f\parallel }_{{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)}\le 1}{sup}}{∥f-S_{\mathsf{\Lambda }}f∥}_{\mathcal{A}\left({\mathbb{T}}^d\right)}.\end{array}$$

Therefore, the operator $S_{\mathsf{\Lambda }}$ is the optimal operator and Corollary 1 is proved.  □

3.2. The Approximation Numbers, Kolmogorov Numbers, and Entropy Numbers of $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)$, for $1\le p,q<\infty$

In this section, we turn to discuss the approximation numbers, Kolmogorov numbers, and entropy numbers of $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to {\mathrm{L}}_q\left({\mathbb{T}}^d\right)$ for $1\le q<\infty$. To obtain a more precise estimation of the approximation numbers, Kolmogorov numbers, and entropy numbers of $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)$, we undertake a comprehensive examination of the parameter q, partitioning it into three distinct scenarios for meticulous analysis: specifically, the cases where $q=2$, $q>2$, and $q<2$.

Case 1. $q=2$:

The subsequent theorem furnishes an estimation of the approximation numbers, Kolmogorov numbers, and entropy numbers of the operator $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to {\mathrm{L}}_q\left({\mathbb{T}}^d\right)$, for $1\le p<\infty$, when $q=2$.

Theorem 2.

Let $\omega =\left\{{\omega }_k\right\}$ be a weight sequence and ${lim}_{{\left|k\right|}_1\to \infty }\left|{\omega }_k\right|=\infty$. Let the sequence ${\left\{{\sigma }_n\right\}}_{n\in N}$ be the non-increasing rearrangement of the sequence ${\left\{1/{\omega }_k\right\}}_{k\in {\mathbb{Z}}^d}$.

(1) If $p=1$ or $2\le p<\infty$, then we have

$$\begin{array}{cc}\hfill a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)& =d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)\hfill \\ & =\left\{\begin{array}{cc}{\displaystyle {sup}_{h\ge n}}{\left(\frac{h-n+1}{{\sum }_{i=1}^h{\sigma }_i^{-2}}\right)}^{1/2},& p=1,\\ {\sigma }_n,& p=2,\\ {\left({\displaystyle {\sum }_{i=n}^{\infty }}{\sigma }_i^r\right)}^{1/r},& 2<p<\infty ,\end{array}\right.\hfill \end{array}$$

where $1/r=1/2-1/p.$

(2) If $1<p<2$, then we derive both the upper and lower bounds of the approximation number and Kolmogorov number as

$$\begin{array}{cc}\hfill {\displaystyle \underset{h\ge n}{sup}}{\left(\frac{h-n+1}{{\sum }_{i=1}^h{\sigma }_i^{-2}}\right)}^{1/2}& \le a_n\left(id:{\mathcal{A}}_{\omega }^p\left(T^d\right)\to L_2\left(T^d\right)\right)\hfill \\ \hfill & =d_n\left(id:{\mathcal{A}}_{\omega }^p\left(T^d\right)\to L_2\left(T^d\right)\right)\le {\sigma }_n.\hfill \end{array}$$

(3) If $1\le p<\infty$, and the sequence $\left\{{\sigma }_n\right\}$ satisfies the condition outlined in Lemma 2(4), then we obtain

$${\epsilon }_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)\asymp n^{1/2-1/p}{\sigma }_n.$$

Proof. 

In order to address the behavior of the approximation numbers, Kolmogorov numbers, and entropy numbers associated with the operator $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to {\mathrm{L}}_2\left({\mathbb{T}}^d\right)$, our approach entails establishing the embedding ${\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\hookrightarrow {\mathrm{L}}_2\left({\mathbb{T}}^d\right)$ as an initial step.

If $1\le p\le 2$, for every $f\in {\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)$, then

$$\parallel \widehat{f}{\left(k\right)\parallel }_{{\ell }_{2\left({\mathbb{Z}}^d\right)}}\le {\parallel \widehat{f}\left(k\right)\parallel }_{{\ell }_p\left({\mathbb{Z}}^d\right)}\le {\parallel {\omega }_k\widehat{f}\left(k\right)\parallel }_{{\ell }_p\left({\mathbb{Z}}^d\right)}<\infty ,$$

which indicates that ${\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\hookrightarrow {\mathrm{L}}_2\left({\mathbb{T}}^d\right)$, for $1\le p\le 2$.

If $p>2$, for every $f\in {\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)$, we have $\left\{\frac{1}{\omega }\right\}\in {\ell }_{1+\frac{1}{p-1}}\left({\mathbb{Z}}^d\right)$. Since ${\ell }_{1+\frac{1}{p-1}}\left({\mathbb{Z}}^d\right)\subseteq {\ell }_{2+\frac{4}{p-2}}\left({\mathbb{Z}}^d\right)$, we have $\left\{\frac{1}{\omega }\right\}\in {\ell }_{2+\frac{4}{p-2}}\left({\mathbb{Z}}^d\right)$. By the Hölder inequality, we obtain $f\in {\mathrm{L}}_2\left({\mathbb{T}}^d\right)$, which shows that ${\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\hookrightarrow L_2\left({\mathbb{T}}^d\right)$, for $p>2$.

Therefore, for $1\le p<\infty$, the embedding of ${\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\hookrightarrow L_2\left({\mathbb{T}}^d\right)$ holds.

Now we turn to consider the diagram as follows

$$\begin{array}{cc}{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\stackrel{id}{⟶}& L_2\left({\mathbb{T}}^d\right)\\ A\downarrow & \uparrow B\\ {\ell }_p\left({\mathbb{Z}}^d\right)\stackrel{D_{\omega }}{⟶}& {\ell }_2\left({\mathbb{Z}}^d\right)\end{array}$$

(8)

where $A,D_{\omega }$, and B are defined in (2) with $\parallel A\parallel =1,\parallel B\parallel =1$. From Lemma 1(4) and the diagram (8) with $id=B{D}_{\omega }A,$ we have

$$\begin{array}{cc}\hfill s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)& \le \parallel A\parallel s_n\left(D_{\omega }:{\ell }_p\left({\mathbb{Z}}^d\right)\to {\ell }_2\left({\mathbb{Z}}^d\right)\right)\parallel B\parallel \hfill \\ \hfill & =s_n\left(D_{\omega }:{\ell }_p\left({\mathbb{Z}}^d\right)\to {\ell }_2\left({\mathbb{Z}}^d\right)\right).\hfill \end{array}$$

On the other hand, we consider the reverse direction diagram as

$$\begin{array}{cc}{\ell }_p\left({\mathbb{Z}}^d\right)\stackrel{D_{\omega }}{⟶}& {\ell }_2\left({\mathbb{Z}}^d\right)\\ A^{-1}\downarrow & \uparrow B^{-1}\\ {\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\stackrel{id}{⟶}& L_2\left({\mathbb{T}}^d\right)\end{array}$$

(9)

Similarly, from Lemma 1(4) and the diagram (9) with $id=B^{-1}{D}_{\omega }{A}^{-1}$ and $∥A^{-1}∥=∥B^{-1}∥=1$, we obtain the lower bound of the approximation numbers, Kolmogorov numbers, and entropy numbers of the embedding of weighted p-Wiener algebra as

$$\begin{array}{cc}\hfill s_n\left(D_{\omega }:{\ell }_p\left({\mathbb{Z}}^d\right)\to {\ell }_2\left({\mathbb{Z}}^d\right)\right)& \le \parallel A^{-1}\parallel s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)\parallel B^{-1}\parallel \hfill \\ \hfill & =s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right).\hfill \end{array}$$

Thus, it is viable to transform the approximation number, Kolmogorov number, and entropy number associated with the mapping $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)$ into their counterparts for the mapping $D_{\omega }:{\ell }_p\left({\mathbb{Z}}^d\right)\to {\ell }_2\left({\mathbb{Z}}^d\right)$. This transformation enables a comprehensive examination and comparison between the numerical measures characterizing the embeddings of these two function spaces. Moreover, since ${\left\{{\sigma }_n\right\}}_{n\in \mathbb{N}}$ is the non-increasing rearrangement of ${(1/{\omega }_k)}_{k\in {\mathbb{Z}}^d}$, we obtain this transformation relationship as

$$\begin{array}{cc}\hfill s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)& =s_n\left(D_{\omega }:{\ell }_p\left({\mathbb{Z}}^d\right)\to {\ell }_2\left({\mathbb{Z}}^d\right)\right)\hfill \\ \hfill & =s_n\left(\sigma :{\ell }_p\left(\mathbb{N}\right)\to {\ell }_2\left(\mathbb{N}\right)\right).\hfill \end{array}$$

(10)

(1) When $p=1$ or $2\le p<\infty$, according to Lemma 2(1), (2), (3) and Equation (10), the approximation number and Kolmogorov number of $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)$ can be estimated as

$$\begin{array}{cc}\hfill a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)& =d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)\hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle {sup}_{h\ge n}}{\left(\frac{h-n+1}{{\sum }_{i=1}^h{\sigma }_i^{-2}}\right)}^{1/2},& p=1,\\ {\sigma }_n,& p=2,\\ {\left({\displaystyle {\sum }_{i=n}^{\infty }}{\sigma }_i^r\right)}^{1/r},& 2<p<\infty ,\end{array}\right.\hfill \end{array}$$

(11)

where $1/r=1/2-1/p.$

(2) When $1<p<2$, it holds that

$${\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right)\hookrightarrow {\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\hookrightarrow {\mathcal{A}}_{\omega }^2\left({\mathbb{T}}^d\right).$$

(12)

For any given functions $f\in {\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right)$ and $g\in {\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)$, it can be ascertained that ${\parallel f\parallel }_{{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)}\le {\parallel f\parallel }_{{\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right)}$, and similarly, ${\parallel g\parallel }_{{\mathcal{A}}_{\omega }^2\left({\mathbb{T}}^d\right)}\le {\parallel g\parallel }_{{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)}$. Then, it can be inferred that the unit balls $B{\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right),B{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)$, and $B{\mathcal{A}}_{\omega }^2\left({\mathbb{T}}^d\right)$ of the function spaces ${\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right)$, ${\mathcal{A}}_{\omega }^2\left({\mathbb{T}}^d\right)$, and ${\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)$, respectively, exhibit the relationship $B{\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right)\subseteq B{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\subseteq B{\mathcal{A}}_{\omega }^2\left({\mathbb{T}}^d\right)$.

Therefore, we can provide an upper bound and a lower bound for the approximation number and Kolmogorov number pertaining to the mapping $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)$ as

$$\begin{array}{cc}\hfill s_n\left(id:{\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)& \le s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)\hfill \\ \hfill & \le s_n\left(id:{\mathcal{A}}_{\omega }^2\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right).\hfill \end{array}$$

(13)

Finally, from

$$\begin{array}{cc}\hfill a_n\left(id:{\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)& =d_n\left(id:{\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)\hfill \\ \hfill & ={\displaystyle \underset{h\ge n}{sup}}{\left(\frac{h-n+1}{{\sum }_{i=1}^h{\sigma }_i^{-2}}\right)}^{1/2},\hfill \end{array}$$

see ([19], Theorem 3.3), Equation (11) and Inequality (13), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \underset{h\ge n}{sup}}{\left(\frac{h-n+1}{{\sum }_{i=1}^h{\sigma }_i^{-2}}\right)}^{1/2}& \le a_n\left(id:{\mathcal{A}}_{\omega }^p\left(T^d\right)\to L_2\left(T^d\right)\right)\hfill \\ \hfill & =d_n\left(id:{\mathcal{A}}_{\omega }^p\left(T^d\right)\to L_2\left(T^d\right)\right)\le {\sigma }_n.\hfill \end{array}$$

(3) When $1\le p<\infty$, from Lemma 2(4) and Equation (10), the entropy number can be estimated as

$${\epsilon }_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)\asymp n^{1/2-1/p}{\sigma }_n.$$

Consequently, the proof of Theorem 2 is hereby established.  □

Case 2. $q\ne 2$:

The subsequent theorems aim to establish the order of the approximation numbers, Kolmogorov numbers, and entropy numbers pertaining to the operator $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to {\mathrm{L}}_q\left({\mathbb{T}}^d\right)$, concerning the cases where $2<q<\infty$ and $1\le q<2$, respectively.

Theorem 3.

Let $\omega =\left\{{\omega }_k\right\}$ be a weight sequence and ${lim}_{{\left|k\right|}_1\to \infty }\left|{\omega }_k\right|=\infty$. Let the sequence ${\left\{{\sigma }_n\right\}}_{n\in N}$ be the non-increasing rearrangement of the sequence ${\left\{1/{\omega }_k\right\}}_{k\in {\mathbb{Z}}^d}$.

For $2<q<\infty$, $1/r_1=1/2-1/p$ and $1/r_2=1-1/p$,

(1) the approximation number and Kolmogorov number pertaining to the operator $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to {\mathrm{L}}_q\left({\mathbb{T}}^d\right)$ can be estimated as

(i) if $p=1$, then

$$\begin{array}{cc}\hfill {\displaystyle \underset{h\ge n}{sup}}{\left(\frac{h-n+1}{{\sum }_{i=1}^h{\sigma }_i^{-2}}\right)}^{1/2}& \le a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\hfill \\ & =d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\le {\sigma }_n;\hfill \end{array}$$

(ii) if $1<p<2$, then

$$\begin{array}{cc}\hfill {\displaystyle \underset{h\ge n}{sup}}{\left(\frac{h-n+1}{{\sum }_{i=1}^h{\sigma }_i^{-2}}\right)}^{1/2}& \le a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\hfill \\ & =d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\le {\left({\displaystyle \sum _{i=n}^{\infty }}{\sigma }_i^{r_2}\right)}^{1/r_2};\hfill \end{array}$$

(iii) if $p=2$, then

$$\begin{array}{cc}\hfill {\sigma }_n& \le a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\hfill \\ & =d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\le {\left({\displaystyle \sum _{i=n}^{\infty }}{\sigma }_i^{r_2}\right)}^{1/r_2};\hfill \end{array}$$

(iv) if $2<p<\infty$, then

$$\begin{array}{cc}\hfill {\left({\displaystyle \sum _{i=n}^{\infty }}{\sigma }_i^{r_1}\right)}^{1/r_1}& \le a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\hfill \\ & =d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\le {\left({\displaystyle \sum _{i=n}^{\infty }}{\sigma }_i^{r_2}\right)}^{1/r_2}.\hfill \end{array}$$

(2) if the sequence $\left\{{\sigma }_n\right\}$ satisfies the condition outlined in Lemma 2(4), then the entropy number pertaining to the operator $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to {\mathrm{L}}_q\left({\mathbb{T}}^d\right)$ can be estimated as

$$n^{1/2-1/p}{\sigma }_n\preccurlyeq {\epsilon }_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\preccurlyeq n^{1-1/p}{\sigma }_n,$$

for $1\le p<\infty$.

Proof. 

For $2<q<\infty$, $1<p<\infty$, from Equation (12), Remark 1, and (Reference [19] Theorem 3.6), we obtain the following chain of embeddings

$${\mathcal{A}}_{\omega }\left({\mathbb{T}}^d\right)\hookrightarrow {\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\hookrightarrow \mathcal{A}\left({\mathbb{T}}^d\right)\hookrightarrow L_{\infty }\left({\mathbb{T}}^d\right)\hookrightarrow L_q\left({\mathbb{T}}^d\right)\hookrightarrow L_2\left({\mathbb{T}}^d\right).$$

Thus, it holds that

$$\begin{array}{cc}\hfill s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)& \le s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\hfill \\ \hfill & \le s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_{\infty }\left({\mathbb{T}}^d\right)\right).\hfill \end{array}$$

(14)

Then, from Equation (14) and Theorem 2, we obtain the lower bound of the approximation numbers, Kolmogorov numbers, and entropy numbers as

$$\begin{array}{cc}\hfill a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)& =d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\hfill \\ & \ge \left\{\begin{array}{cc}{\displaystyle {sup}_{h\ge n}}{\left(\frac{h-n+1}{{\sum }_{i=1}^h{\sigma }_i^{-2}}\right)}^{1/2},& 1\le p<2,\\ {\sigma }_n,& p=2,\\ {\left({\displaystyle {\sum }_{i=n}^{\infty }}{\sigma }_i^{r_1}\right)}^{1/r_1},& 2<p<\infty ,\end{array}\right.\hfill \end{array}$$

and

$${\epsilon }_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\succcurlyeq n^{1/2-1/p}{\sigma }_n.$$

For the upper bound of Theorem 3, from Equation (14), we have

$$\begin{array}{cc}\hfill s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)& \le s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_{\infty }\left({\mathbb{T}}^d\right)\right)\hfill \\ \hfill & \le \parallel id:\mathcal{A}\left({\mathbb{T}}^d\right)\to L_{\infty }\left({\mathbb{T}}^d\right)\parallel ·s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right)\hfill \\ \hfill & =s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right).\hfill \end{array}$$

(15)

Then, based on the result of Theorem 1, Reference [19], and Equation (15), we obtain the upper bound of the approximation numbers, Kolmogorov numbers, and entropy numbers as

$$\begin{array}{cc}\hfill a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)& =d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\hfill \\ & \le \left\{\begin{array}{cc}{\sigma }_n,& p=1,\\ {\left({\displaystyle {\sum }_{i=n}^{\infty }}{\sigma }_i^{r_2}\right)}^{1/r_2},& 1<p<\infty ,\end{array}\right.\hfill \end{array}$$

and

$${\epsilon }_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\preccurlyeq n^{1-1/p}{\sigma }_n.$$

Finally, we obtain the desired results of Theorem 3.  □

Theorem 4.

Let $\omega =\left\{{\omega }_k\right\}$ be a weight sequence and ${lim}_{{\left|k\right|}_1\to \infty }\left|{\omega }_k\right|=\infty$. Let the sequence ${\left\{{\sigma }_n\right\}}_{n\in \mathbb{N}}$ be the non-increasing rearrangement of the sequence ${\left\{1/{\omega }_k\right\}}_{k\in {\mathbb{Z}}^d}$. Then, for $1\le q<2$, $1/q+1/q^{\prime }=1$, $r_1=1/2-1/p$, $r_2=1/q^{\prime }-1/p$, and the sequence $\left\{{\sigma }_n\right\}$ for estimating the entropy number satisfies the condition outlined in Lemma 2(4),

(1) the upper bound of the approximation number, Kolmogorov number, and entropy number pertaining to the operator $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to {\mathrm{L}}_q\left({\mathbb{T}}^d\right)$ can be estimated as

$$\begin{array}{cc}\hfill a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)& =d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\hfill \\ & \le \left\{\begin{array}{cc}{\displaystyle {sup}_{h\ge n}}{\left(\frac{h-n+1}{{\sum }_{i=1}^h{\sigma }_i^{-2}}\right)}^{1/2},& p=1,\\ {\sigma }_n,& 1<p\le 2,\\ {\left({\displaystyle {\sum }_{i=n}^{\infty }}{\sigma }_i^{r_1}\right)}^{1/r_1},& 2<p<\infty ,\end{array}\right.\hfill \end{array}$$

and

$${\epsilon }_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\preccurlyeq n^{1/2-1/p}{\sigma }_n.$$

(2) the lower bound of the approximation number, Kolmogorov number, and entropy number pertaining to the operator $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to {\mathrm{L}}_q\left({\mathbb{T}}^d\right)$ can be estimated as

$$\begin{array}{cc}\hfill a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)& =d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\hfill \\ & \ge \left\{\begin{array}{cc}{\sigma }_n,& 2<p=q^{\prime }<\infty ,\\ {\left({\displaystyle {\sum }_{i=n}^{\infty }}{\sigma }_i^{r_2}\right)}^{1/r_2},& 2<q^{\prime }<p<\infty ,\end{array}\right.\hfill \end{array}$$

and

$${\epsilon }_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\succcurlyeq n^{1/q^{\prime }-1/p}{\sigma }_n.$$

Proof. 

For $1\le q<2$, we consider the following chain of embedding

$$L_2\left({\mathbb{T}}^d\right)\hookrightarrow L_q\left({\mathbb{T}}^d\right).$$

When $1\le p<\infty$, we obtain

$$s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\le s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right).$$

(16)

From the results of Theorem 2 and Equation (16), we can estimate the upper bound of the approximation number, Kolmogorov number, and entropy number as

$$\begin{array}{cc}\hfill a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)& =d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\hfill \\ & \le \left\{\begin{array}{cc}{\displaystyle {sup}_{h\ge n}}{\left(\frac{h-n+1}{{\sum }_{i=1}^h{\sigma }_i^{-2}}\right)}^{1/2},& p=1,\\ {\sigma }_n,& 1<p\le 2,\\ {\left({\displaystyle {\sum }_{i=n}^{\infty }}{\sigma }_i^{r_1}\right)}^{1/r_1},& 2<p<\infty ,\end{array}\right.\hfill \end{array}$$

and

$${\epsilon }_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\preccurlyeq n^{1/2-1/p}{\sigma }_n.$$

Subsequently, we turn to estimate the lower bound of Theorem 4.

When $1\le q<2$ and $q^{\prime }$ denotes the conjugate index of q, satisfying $1/q^{\prime }+1/q=1$, it is obvious that $1\le q<2<q^{\prime }\le \infty$, and $\parallel \left\{\widehat{f}\left(k\right)\right\}{\parallel }_{{\ell }_{q^{\prime }}\left({\mathbb{Z}}^d\right)}\le {\parallel \left\{\widehat{f}\left(k\right)\right\}\parallel }_{{\ell }_q\left({\mathbb{Z}}^d\right)}$ holds.

For every $f\in L_q\left({\mathbb{T}}^d\right)$, the Hausdorff–Young inequality [see [24], Theorem 3.2.2] establishes that the sequence of Fourier coefficients of f belongs to the sequence space ${\ell }_{q^{\prime }}\left({\mathbb{Z}}^d\right)$, and it further satisfies the inequality:

$$\parallel \left\{\widehat{f}\left(k\right)\right\}\parallel {}_{{\ell }_{q^{\prime }}\left({\mathbb{Z}}^d\right)}\le {\parallel f\parallel }_{L_q\left({\mathbb{T}}^d\right)}.$$

(17)

We consider the following diagram

$$\begin{array}{cc}{\ell }_p\left({\mathbb{Z}}^d\right)\stackrel{D_{\omega }}{⟶}& {\ell }_{q^{\prime }}\left({\mathbb{Z}}^d\right)\\ A^{-1}\downarrow & \uparrow B_1^{-1}\\ {\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\stackrel{id}{⟶}& L_q\left({\mathbb{T}}^d\right)\end{array}$$

(18)

where A, $D_{\omega }$ are defined as in (2) with $\parallel A\parallel =\parallel A^{-1}\parallel =1$, and $B_1$ is defined as

$$\left(B_1\xi \right)\left(x\right):={\displaystyle \sum _{k\in {\mathbb{Z}}^d}}{\xi }_k{e}^{ikx},x\in {\mathbb{T}}^d.$$

Thus, from Equation (17), we obtain $\parallel B_1^{-1}\parallel \le 1$.

From Lemma 1 and $id=B_1^{-1}{D}_{\omega }{A}^{-1}$ with $\parallel A^{-1}\parallel =\parallel B_1^{-1}\parallel =1$, we have

$$\begin{array}{cc}{s}_n\left(D_{\omega }:{\ell }_p\left({\mathbb{Z}}^d\right)\to {\ell }_{q^{\prime }}\left({\mathbb{Z}}^d\right)\right)& \le \parallel A^{-1}\parallel s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\parallel B_1^{-1}\parallel \\ & \le s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right).\end{array}$$

(19)

Finally, when $2<q^{\prime }<p<\infty$, from Lemma 2(1), Lemma 2(2), and Equation (19), we obtain

$$a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)=d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\ge {\left({\displaystyle \sum _{i=n}^{\infty }}{\sigma }_i^{r_2}\right)}^{1/r_2}.$$

When $2<q^{\prime }=p<\infty$, from Lemma 2(2) and Equation (19), we obtain

$$a_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)=d_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\ge {\sigma }_n.$$

Therefore, results of Theorem 4 are obtained.  □

Remark 3.

Theorems 3 and 4 present upper and lower bound estimates for the approximation number, Kolmogorov number, and entropy number associated with the operator $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to {\mathrm{L}}_q\left({\mathbb{T}}^d\right)$ as for $2<q<\infty ,$

$$s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right)\le s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\le s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)\right),$$

and for $1\le q<2,$

$$s_n\left(D_{\omega }:{\ell }_p\left({\mathbb{Z}}^d\right)\to {\ell }_{q^{\prime }}\left({\mathbb{Z}}^d\right)\right)\le s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)\right)\le s_n\left(id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_2\left({\mathbb{T}}^d\right)\right).$$

Therefore, for $q\ne 2$, we can not estimate the exact order of the approximation number, Kolmogorov number, and entropy number pertaining to the operator $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)$, and alternative estimation methods are required to address the exact estimates.

4. Conclusions

Approximation numbers, Kolmogorov numbers, and entropy numbers, often referred to as s-numbers or pseudo s-numbers, are important concepts in the field of functional analysis. They are indispensable in various fields such as mathematics, numerical analysis, and scientific computing. In this paper, we propose the concept of the weighted p-Wiener algebra and examine the asymptotic behavior of the approximation number, Kolmogorov number, and entropy number associated with the embeddings $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to \mathcal{A}\left({\mathbb{T}}^d\right)$ and $id:{\mathcal{A}}_{\omega }^p\left({\mathbb{T}}^d\right)\to L_q\left({\mathbb{T}}^d\right)$, where $1\le p,q<\infty$.

In the context of mathematical analysis and machine learning, approximation numbers, Kolmogorov numbers, and entropy numbers serve as fundamental theoretical tools that facilitate a more profound comprehension of approximation and complexity in mathematical entities. The notion of s-numbers is intrinsically linked to sparsity and, consequently, holds relevance in the investigation of sparse representations within the realm of machine learning, encompassing domains like compressed sensing and sparse coding. Additionally, the application of entropy numbers in data compression scenarios assumes significance given their pivotal role in machine learning algorithms. These numbers furnish a quantification of the efficacy with which a dataset can be compressed, thereby gaining significance when handling voluminous datasets or devising streamlined data representation methodologies. The implications of these theoretical constructs extend to diverse aspects of machine learning, including algorithm design, error analysis, regularization, and comprehension of the generalization performance exhibited by various machine learning methodologies. As part of our future endeavors, we shall delve further into exploring these aforementioned implications and their practical applications within the machine learning domain.