Approximation Characteristics of Weighted Sobolev Spaces on Sphere in Different Settings

Abstract

This article primarily examines the approximation properties of a weighted Sobolev space $W_{2,\kappa }^r$ defined on a sphere ${\mathbb{S}}^{d-1}$ equipped with Gaussian measures. Specifically, this study focuses on both the average case and probabilistic case settings. The exact asymptotic orders of the Gel’fand n-width and the linear n-width of $W_{2,\kappa }^r$ on ${\mathbb{S}}^{d-1}$ are derived for these settings, providing a comprehensive understanding of their approximation characteristics.

1. Introduction

With the advancement of science and technology and the development of societal needs, computer technology has penetrated various fields such as industry, daily life, and the economy, becoming a primary computational tool. However, as a computational tool, computer technology has inherent limitations, namely limited computational resources. Even today, for vast amounts of data, direct processing by computers is exceedingly slow, and the associated costs are undoubtedly enormous. Therefore, processing data through different algorithms becomes crucial, and finding the algorithm with the lowest cost is even more important. This has driven the development of the field of computational complexity. Computational complexity can be understood as the study of finding the minimal computational cost required to solve a problem within a reasonable error margin (see [1]).

To better understand and address computational complexity, researchers have proposed various computational models and settings, with the most common ones being the worst case, average case, and probabilistic settings. These three options provide different perspectives for analyzing and optimizing algorithm performance.

Width is an important concept in the theory of function approximation, and it plays a vital role in the study of computational complexity (see [2]). Width is commonly used to measure the optimal or minimum error achievable when approximating a known function in a function space using a finite number of information points. Essentially, studying the width is equivalent to studying computational complexity. In different settings, the width provides a quantifiable measure of the complexity of a problem. Therefore, width is not only a theoretical tool for quantifying the limits of algorithmic approximation capabilities but also closely linked to computational complexity. By studying the width under different settings, we can gain a clearer understanding of a problem’s complexity and design more efficient algorithms to solve these problems.

This paper primarily investigates the approximation problem of weighted Sobolev spaces on a sphere ${\mathbb{S}}^{d-1}$, considering both probabilistic and average case settings. The foundational concepts of the n-width will be introduced and reviewed first.

Definition 1 

([2,3]). Let W be a bounded subset of a normed linear space $(X,\parallel ·\parallel )$. Then, the Kolmogorov n-width, the linear n-width, and the Gel’fand n-width of W in X are respectively defined as follows:

$$\begin{array}{cc}& d_n\left(W,X\right):=\underset{F_n}{inf}\underset{x\in W}{sup}\underset{y\in F_n}{inf}∥x-y∥,\hfill \\ & {\lambda }_n\left(W,X\right):=\underset{{\phi }_n}{inf}\underset{x\in W}{sup}∥x-{\phi }_{n}x∥,\hfill \\ & d^n\left(W,X\right):=\underset{L^n}{inf}\underset{x\in W\cap L^n}{sup}∥x∥,\hfill \end{array}$$

where $F_n$ ranges over all linear subspaces of X with a dimension of n at most, ${\phi }_n$ denotes all linear operators from X to X with a rank of n at most, and $L^n$ refers to all linear subspaces of X with a codimension no greater than n.

Clearly, the above definition characterizes the best approximation errors for the most difficult elements in W when approximated by n-dimensional subspaces, linear operators with a rank n, and subspaces of a codimension n. If these errors fall within a given tolerance, then this implies that all elements of W are considered acceptable.

We now shift our focus to probabilistic and average case settings. In these scenarios, it becomes essential to establish certain measurability assumptions. Let W be equipped with a Borel $\sigma$ algebra $\mathcal{B}$, consisting of the open subsets of W, and a probability measure v defined on $\mathcal{B}$. Specifically, v is a nonnegative, $\sigma$-additive measure on $\mathcal{B}$ which satisfies $v\left(W\right)=1$.

Definition 2 

([4,5,6]). Let W, X, and v be defined as above, and let δ be a number in the range of $[0,1)$. The probabilistic Kolmogorov $(n,\delta )$-width, linear $(n,\delta )$-width, and Gel’fand $(n,\delta )$-width of W in X are respectively defined as follows:

$$\begin{array}{cc}& d_{n,\delta }(W,v,X):=\underset{G_{\delta }}{inf}{d}_n(W\setminus G_{\delta },X),\hfill \\ & {\lambda }_{n,\delta }(W,v,X):=\underset{G_{\delta }}{inf}{\lambda }_n(W\setminus G_{\delta },X),\hfill \\ & d_{\delta }^n(W,v,X):=\underset{G_{\delta }}{inf}{d}^n(W\setminus G_{\delta },X),\hfill \end{array}$$

where $G_{\delta }$ represents subsets within $\mathcal{B}$ such that $v\left(G_{\delta }\right)\le \delta$.

By comparing the definitions in the probabilistic setting with those in the worst case setting, it is easy to see that by removing a set in $\mathcal{B}$ with a measure not exceeding $\delta$, the errors caused by extreme elements could be significantly reduced in certain cases. Compared with the worst case setting, this approach better reflects the intrinsic structure of the classes. On the other hand, from the perspective of computational complexity, the probabilistic approach can, in some cases, greatly reduce the model complexity, thereby lowering the required cost.

Remark 1. 

The probabilistic Gel’fand n-width was first introduced in [4]. The definition in this paper indicates that the excluded measure set $G_{\delta }$ must satisfy additional measure conditions, which are detailed in the same paper. However, as mentioned in [7] (Remark 1.2), the proof in this article only requires some basic results for the probabilistic Gel’fand width. Consequently, the additional measurement conditions are similarly excluded in this context.

Definition 3 

([6,7,8]). Let W, X, and v be defined as above. Then the p-average Kolmogorov n-width, p-average linear n-width, and p-average Gel’fand n-width are respectively defined as follows:

$$\begin{array}{cc}& d_n^{\left(a\right)}{(W,v,X)}_p:=\underset{F_n}{inf}{\left({\int }_W(\underset{y\in F_n}{inf}{\parallel x-y\parallel )}^p\mathrm{d}v\left(x\right)\right)}^{{\displaystyle \frac{1}{p}}},\hfill \\ & {\lambda }_n^{\left(a\right)}{(W,v,X)}_p:=\underset{{\phi }_n}{inf}{\left({\int }_W{\parallel x-{\phi }_{n}x\parallel }^p\mathrm{d}v\left(x\right)\right)}^{{\displaystyle \frac{1}{p}}},\hfill \\ & d_{\left(a\right)}^n{(W,v,X)}_p:=\underset{L^n}{inf}{\left({\int }_{W\cap L^n}{\parallel x\parallel }^p\mathrm{d}v\left(x\right)\right)}^{{\displaystyle \frac{1}{p}}},\hfill \end{array}$$

where $0<p<\infty$, $F_n$ ranges over all linear subspaces of X with dim $X\le n$, ${\phi }_n$ denotes any linear operator from X to X with a rank not exceeding n, and $L^n$ refers to all linear subspaces of X with a codimension no greater than n.

The Kolmogorov width and linear width in the average case setting have been studied for over 20 years. In contrast, the Gel’fand width in the average case setting was only recently defined in [7], and there is limited research on this topic. These concepts are used to characterize the ability to optimally approximate the majority of the elements in the function class under consideration using subspaces or linear operators. Obviously, this method better reflects the actual situation of the class.

The paper employs the following notation. For two nonnegative functions $\gamma \left(y\right)$ and $\beta \left(y\right)$, where $y\in D$, if there exists a positive constant $c_1$ independent of y such that $\gamma \left(y\right)\le c_1\beta \left(y\right)$ holds, then this relationship is denoted by $\gamma \left(y\right)\ll \beta \left(y\right)$. Conversely, if there exists a positive constant $c_2$ independent of y satisfying $\gamma \left(y\right)\ge c_2\beta \left(y\right)$, then this relationship is denoted by $\gamma \left(y\right)\gg \beta \left(y\right)$. If $\gamma \left(y\right)\ll \beta \left(y\right)$ and $\gamma \left(y\right)\gg \beta \left(y\right)$, then this relationship is denoted by $\gamma \left(y\right)\asymp \beta \left(y\right)$.

Many papers have studied the problem of n-widths in the worst case, probabilistic, and average case settings, yielding numerous remarkable results. Here, we review some of the known findings:

(1)

Consider a classical univariate Sobolev space $W_2^r$ equipped with a Gaussian measure $\mu$ and satisfying the conditions $\rho >1$, $r>{\displaystyle \frac{1}{2}}$, and $\delta \in (0,{\displaystyle \frac{1}{2}}]$, where $\rho >1$ ensures a well-defined measure $\mu$. Maiorov in [5,6] proved that

$$\begin{array}{cc}\hfill d_{n,\delta }\left(W_2^r,\mu ,L_q\right)& \asymp n^{-\left(r+{\displaystyle \frac{\rho -1}{2}}\right)}\sqrt{1+{\displaystyle \frac{1}{n}}ln{\displaystyle \frac{1}{\delta }}},1\le q\le \infty ,\hfill \\ \hfill d_n^{\left(a\right)}{\left(W_2^r,\mu ,L_q\right)}_p& \asymp n^{-\left(r+{\displaystyle \frac{\rho -1}{2}}\right)},1\le q\le \infty ,0<p<\infty ,\hfill \\ \hfill {\lambda }_{n,\delta }\left(W_2^r,\mu ,L_q\right)& \asymp n^{-\left(r+{\displaystyle \frac{\rho -1}{2}}\right)}\left(1+n^{-{\displaystyle \frac{1}{q}}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}\right),2\le q<\infty .\hfill \end{array}$$

Fang and Ye extended Maiorov’s results in [8] and obtained

$$\begin{array}{cc}\hfill {\lambda }_{n,\delta }\left(W_2^r,\mu ,L_q\right)& \asymp n^{-\left(r+{\displaystyle \frac{\rho -1}{2}}\right)}\left(1+n^{-{\displaystyle \frac{1}{2}}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}\right),1\le q<2,\hfill \\ \hfill {\lambda }_n^{\left(a\right)}{\left(W_2^r,\mu ,L_q\right)}_p& \asymp n^{-\left(r+{\displaystyle \frac{\rho -1}{2}}\right)},1\le q<\infty ,0<p<\infty .\hfill \end{array}$$

However, no results were available for Gel’fand widths until recently, when Tan et al. obtained the following result in [4]:

$$d_{\delta }^n(W_2^r,\mu ,L_q)\asymp \left\{\begin{array}{cc}{n}^{-\left(r+{\displaystyle \frac{\rho -1}{2}}\right)}\sqrt{1+{\displaystyle \frac{1}{n}}ln{\displaystyle \frac{1}{\delta }}},\hfill & 1\le q<2,\hfill \\ n^{-\left(r+{\displaystyle \frac{\rho -1}{2}}\right)}\left(1+n^{-{\displaystyle \frac{1}{q}}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}\right),\hfill & 2\le q<\infty .\hfill \end{array}\right.$$

Subsequently, Liu et al. obtained the following result in [7]:

$$d_{\left(a\right)}^n{\left(W_2^r,\mu ,L_q\right)}_p\asymp n^{-\left(r+{\displaystyle \frac{\rho -1}{2}}\right)},1\le q<\infty ,0<p<\infty .$$

(2)

Consider a multivariate Sobolev space $M{W}_2^r$ with mixed derivatives equipped with a measure $\mu$ and satisfying the conditions $\rho >1$, $r=(r_1,\dots ,r_d)$, ${\displaystyle \frac{1}{2}}<r_1=\cdots =r_v<r_{v+1}\le \cdots \le r_d$, and $\delta \in (0,{\displaystyle \frac{1}{2}}]$. Similarly, $\rho >1$ ensures that the Gaussian measure $\mu$ is well defined. The authors in [7,9,10] obtained the following for $1<q<2$:

$$\begin{array}{cc}\hfill & T_{n,\delta }(M{W}_2^r,\mu ,L_q)\asymp {\left(n^{-1}{ln}^{v-1}n\right)}^{r_1+{\displaystyle \frac{\rho -1}{2}}}\left({ln}^{{\displaystyle \frac{v-1}{2}}}n\right)\sqrt{1+{\displaystyle \frac{1}{n}}ln\left({\displaystyle \frac{1}{\delta }}\right)},\hfill \\ \hfill & T_n^{\left(a\right)}{(M{W}_2^r,\mu ,L_q)}_p\asymp {\left(n^{-1}{ln}^{v-1}n\right)}^{r_1+{\displaystyle \frac{\rho -1}{2}}}\left({ln}^{{\displaystyle \frac{v-1}{2}}}n\right),\hfill \end{array}$$

(1)

where $T_{n,\delta }$ represents $d_{n,\delta }$, ${\lambda }_{n,\delta }$, and $d_{\delta }^n$, $T_n^{\left(a\right)}$ represents $d_n^{\left(a\right)}$, ${\lambda }_n^{\left(a\right)}$, and $d_{\left(a\right)}^n$, and $p,\rho$ are defined as mentioned above. Additionally, for the case $2\le q<\infty$, Equation (1) still holds for the Kolmogorov width. However, as noted in [7,10], the linear width and Gel’fand width are somewhat different:

$$\begin{array}{cc}& {ln}^{{\displaystyle \frac{v-1}{q}}}n\ll {\displaystyle \frac{Y_{n,\delta }(M{W}_2^r,\mu ,L_q)}{{\left(n^{-1}{ln}^{v-1}n\right)}^{r_1+\frac{\rho -1}{2}}\left(1+n^{-\frac{1}{q}}\sqrt{ln\frac{1}{\delta }}\right)}}\ll {ln}^{{\displaystyle \frac{v-1}{2}}}n,\hfill \\ & {ln}^{{\displaystyle \frac{v-1}{q}}}n\ll {\displaystyle \frac{Y_n^{\left(a\right)}(M{W}_2^r,\mu ,L_q)}{{\left(n^{-1}{ln}^{v-1}n\right)}^{r_1+\frac{\rho -1}{2}}}}\ll {ln}^{{\displaystyle \frac{v-1}{2}}}n,\hfill \end{array}$$

where $Y_{n,\delta }$ represents ${\lambda }_{n,\delta }$ and $d_{\delta }^n$, $Y_n^{\left(a\right)}$ represents ${\lambda }_n^{\left(a\right)}$ and $d_{\left(a\right)}^n$, and $p,\rho$ are defined as mentioned above.

(3)

The weighted Sobolev spaces $W_{p,\mu }^r$ on the ball ${\mathbb{B}}^d$ and $W_{p,\kappa }^r$ on the sphere ${\mathbb{S}}^{d-1}$ have been extensively studied in the literature. Wang and Huang obtained the asymptotic order of the linear width and Kolmogorov width of the weighted Sobolev space $W_{p,\mu }^r$ on the ball under the deterministic setting in [11]. Huang and Wang obtained the asymptotic order of the linear width, the Gel’fand width, and the Kolmogorov width of the weighted Sobolev space $W_{p,\kappa }^r$ on the sphere under the deterministic setting in [12]. Due to the length of the results, they are not listed here. Additionally, Wang obtained the linear width for the weighted Sobolev space $W_{p,\mu }^r$ on the ball in the probabilistic and average case settings (see [13]) for $r>(d+2\mu ){({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}_{+}$ and $\rho =r+{\displaystyle \frac{s}{2}}>{\displaystyle \frac{d}{2}}+2\mu d{({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}_{+},$ $s>d$, where $\delta \in (0,{\displaystyle \frac{1}{2}}]$, $1\le q\le \infty$, and $0<p<\infty$:

$${\lambda }_{n,\delta }(W_{2,\mu }^r,\nu ,L_{q,\mu })\asymp \left\{\begin{array}{cc}{n}^{-{\displaystyle \frac{\rho }{d}}+{\displaystyle \frac{1}{2}}}\left(1+n^{-min\left\{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{q}}\right\}}{\left(ln\left({\displaystyle \frac{1}{\delta }}\right)\right)}^{{\displaystyle \frac{1}{2}}}\right),\hfill & 1\le q<\infty ,\hfill \\ n^{-{\displaystyle \frac{\rho }{d}}+{\displaystyle \frac{1}{2}}}{\left(ln\left({\displaystyle \frac{n}{\delta }}\right)\right)}^{{\displaystyle \frac{1}{2}}},\hfill & q=\infty ,\hfill \end{array}\right.$$

and

$${\lambda }_n^{\left(a\right)}{(W_{2,\mu }^r,\nu ,L_{q,\mu })}_p\asymp \left\{\begin{array}{cc}{n}^{-{\displaystyle \frac{\rho }{d}}+{\displaystyle \frac{1}{2}}},\hfill & 1\le q<\infty ,\hfill \\ n^{-{\displaystyle \frac{\rho }{d}}+{\displaystyle \frac{1}{2}}}\sqrt{ln\left(en\right)},\hfill & q=\infty ,\hfill \end{array}\right.$$

where v represents the Gaussian measure on $W_{2,\mu }^r$. Note that $s>d$ is required to ensure that the Gaussian measure v is well defined. However, the width of weighted Sobolev spaces on a sphere in probabilistic and average case settings has thus far been inconclusive.

Aside from the papers mentioned above, there are many excellent papers on this topic, such as [3,14,15]. However, due to space limitations, they will not be listed here.

Inspired by these studies, this paper will investigate the Gel’fand and linear n-widths of the weighted Sobolev space $W_{2,\kappa }^r$ on the sphere ${\mathbb{S}}^{d-1}$ in the probabilistic and average case settings.

The structure of this paper is as follows. In Section 2, we introduce the primary results of this article, along with essential notations and the Gaussian measure v defined in the weighted Sobolev space, including its key properties. Section 3 presents several lemmas which are fundamental to the proof of the main theorems. Section 4 will present the discretization theorem, which primarily transforms the widths between different function classes, thereby simplifying the computation process. In Section 5, we integrate the discretization theorems to provide the proof of the main theorems. In Section 6, a summary of and outlook for this paper are provided.

2. Preliminaries and Main Results

Let ${\mathbb{S}}^{d-1}:=\{x=(x_1,\dots ,x_d)\in {\mathbb{R}}^d:{\sum }_{j=1}^d{x}_j^2=1\}$ denote the unit sphere in ${\mathbb{R}}^d$, endowed with the standard rotation-invariant measure $\mathrm{d}\sigma \left(x\right)$. Given $\kappa =({\kappa }_1,\dots ,{\kappa }_d)\in {\mathbb{R}}^d$ with $\underset{1\le j\le d}{min}{\kappa }_j>0$ and $\left|\kappa \right|:={\kappa }_1+\cdots +{\kappa }_d$, let

$$W_{\kappa }\left(x\right):=a_k\prod _{i=1}^d{\left|x_i\right|}^{2{\kappa }_i}$$

(2)

denote the Jacobi weight on ${\mathbb{S}}^{d-1}$ and $a_k={\left({\int }_{{\mathbb{S}}^{d-1}}({\displaystyle \prod _{i=1}^d}|x_i{|}^{2{\kappa }_i})\mathrm{d}\sigma \left(x\right)\right)}^{-1}.$

The space $L_{p,\kappa }\equiv L_p\left({\mathbb{S}}^{d-1},W_{\kappa }\left(x\right)\mathrm{d}\sigma \left(x\right)\right)$ for $1\le p<\infty$ is defined as the set of measurable functions on ${\mathbb{S}}^{d-1}$ for which the following norm is finite:

$${\parallel f\parallel }_{p,\kappa }:={\left({\int }_{{\mathbb{S}}^{d-1}}{\left|f\left(x\right)\right|}^p{W}_{\kappa }\left(x\right)\mathrm{d}\sigma \left(x\right)\right)}^{{\displaystyle \frac{1}{p}}},1\le p<\infty .$$

For $p=\infty$, we replace the space $L_{\infty ,\kappa }$ with $C\left({\mathbb{S}}^{d-1}\right)$, which is the space of continuous functions on the sphere ${\mathbb{S}}^{d-1}$ endowed with the uniform norm. When $p=2$, the space $L_{2,\kappa }$ is endowed with the inner product

$$\langle f,g\rangle :={\int }_{{\mathbb{S}}^{d-1}}f\left(x\right)g\left(x\right)W_{\kappa }\left(x\right)\mathrm{d}\sigma \left(x\right),$$

which clearly makes it a Hilbert space.

The Dunkl operators corresponding to the weight function given in Equation (2) are expressed as follows:

$$D_{j}f\left(x\right):={\partial }_{j}f\left(x\right)+{\kappa }_j{\displaystyle \frac{f\left(x\right)-f\left({\sigma }_{j}x\right)}{x_j}},1\le j\le d,$$

where ${\sigma }_{j}x=(x_1,...,-x_j,..,x_d)$. A spherical h-harmonic P of a degree n on ${\mathbb{S}}^{d-1}$ is obtained by restricting a homogeneous polynomial P of a degree n to the sphere such that ${\Delta }_{h}P=0$. Here, ${\Delta }_h$ denotes the h-Laplacian, defined as ${\Delta }_h=D_1^2+\cdots +D_d^2$, which plays a role analogous to that of the standard Laplacian. For further details, refer to [16,17]. The notation ${\Pi }_n^d$ denotes the space of all polynomials in d variables with a degree of n at most. Likewise, ${\mathcal{H}}_n^d$ represents the subspace consisting of n-degree polynomials which are orthogonal to those of a lower degree in $L_{2,\kappa }$. Thus, under classical Hilbert space theory, the following relationship is clearly valid:

$$L_{2,\kappa }=\underset{n=0}{\overset{\infty }{⨁}}{\mathcal{H}}_n^d,{\Pi }_n^d=\underset{k=0}{\overset{n}{⨁}}{\mathcal{H}}_k^d.$$

Moreover, it is known that

$$dim{\mathcal{H}}_n^d\asymp n^{d-2},dim{\Pi }_n^d\asymp n^{d-1}.$$

Let

$$\{{\varphi }_{nk}\mid k=1,\dots ,a_n^d:=dim{\mathcal{H}}_n^d\}$$

be a fixed orthonormal basis for ${\mathcal{H}}_n^d$. Then,

$$\{{\varphi }_{nk}\mid k=1,\dots ,a_n^d,n=0,1,2,\dots \}$$

is an orthonormal basis for $L_{2,\kappa }$.

We use $pro{j}_n$ to represent the orthogonal projection from $L_{2,\kappa }$ onto ${\mathcal{H}}_n^d$ and $S_n$ to denote the orthogonal projection from $L_{2,\kappa }$ onto ${\Pi }_n^d$; that is, for any $f\in L_{2,\kappa }$, we have

$$S_n\left(f\right):=\sum _{k=0}^{n}pro{j}_k\left(f\right),pro{j}_n\left(f\right)\left(x\right):=\sum _{k=1}^{a_n^d}\langle {\varphi }_{nk},f\rangle {\varphi }_{nk}\left(x\right).$$

(3)

Remark 2. 

The operator $S_n$ is also known as the Fourier partial summation operator, and $pro{j}_n$ has the following representation:

$$pro{j}_n\left(f\right)\left(x\right)={\int }_{{\mathbb{S}}^{d-1}}f\left(y\right)P_n(x,y)W_{\kappa }\left(y\right)\mathrm{d}\sigma \left(y\right),$$

where $P_n(x,y)={\sum }_{k=1}^{a_n^d}{\varphi }_{nk}\left(x\right){\varphi }_{nk}\left(y\right)$ is the reproducing kernel for ${\mathcal{H}}_n^d$. Moreover, the reproducing kernel $P_n(x,y)$ has the explicit representation, which can be found in [12,18].

It is well known that spherical h-harmonics are eigenfunctions of the operator ${\Delta }_{h,0}$, meaning

$${\Delta }_{h,0}y=-n(n+2{\lambda }_{\kappa })y,y\in {\mathcal{H}}_n^d,$$

$\mathrm{where}{\lambda }_{\kappa }={\displaystyle \frac{d-2}{2}}+\left|\kappa \right|,\left|\kappa \right|={\sum }_{i=1}^d{\kappa }_i,$${\Delta }_{h,0}$ is also referred to as the Laplace–Beltrami operator.

Given $r>0$, we define the fractional power ${(-D_k)}^{{\displaystyle \frac{r}{2}}}$ by

$${(-D_{\kappa })}^{{\displaystyle \frac{r}{2}}}\left(f\right):=\sum _{n=0}^{\infty }{\left(n(n+2{\lambda }_{\kappa })\right)}^{{\displaystyle \frac{r}{2}}}pro{j}_n\left(f\right)=\sum _{n=0}^{\infty }{\zeta }_n^{{\displaystyle \frac{r}{2}}}pro{j}_n\left(f\right)$$

in a distributional sense. We define $f^{\left(r\right)}:={(-D_{\kappa })}^{{\displaystyle \frac{r}{2}}}\left(f\right)$ to represent the derivative of order r of the distribution f.

The following introduces the spaces primarily studied in this paper. For $r>0$, the weighted Sobolev space $W_{2,\kappa }^r\equiv W_{2,\kappa }^r\left({\mathbb{S}}^{d-1}\right)$ is defined by

$$W_{2,\kappa }^r:=\left\{f\in L_{2,\kappa }|{\int }_{{\mathbb{S}}^{d-1}}f\left(x\right)W_{\kappa }\left(x\right)\mathrm{d}\sigma \left(x\right)=0,{\parallel f\parallel }_{W_{2,\kappa }^r}^2:=\langle f^{\left(r\right)},f^{\left(r\right)}\rangle <\infty \right\}$$

with an inner product

$${\langle f,g\rangle }_r:=\langle f^{\left(r\right)},g^{\left(r\right)}\rangle .$$

It is obvious that this space is a Hilbert space.

This paper primarily studies the width problem of $W_{2,\kappa }^r$ in $L_{q,\kappa }$, where $1\le q\le \infty$. In order to ensure that this problem is well defined, we need $W_{2,\kappa }^r$ to be continuously embedded in $L_{q,\kappa }$. According to Lemma 2, when $r>(d-1+2s_{\kappa }){({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}_{+}$, the embedding is guaranteed, which ensures that the definition is meaningful.

Next, we endow the space $W_{2,\kappa }^r$ with a Gaussian measure v having zero mean, and its correlation operator $C_v$ has eigenfunctions ${\varphi }_{lk},$ where $k=1,\dots ,a_l^d$, and $l=1,2,\dots$, and eigenvalues

$$v_l={\zeta }_l^{-{\displaystyle \frac{s}{2}}},s>d-1,$$

In other words, we have

$$C_v{\varphi }_{lk}={\zeta }_l^{-{\displaystyle \frac{s}{2}}}{\varphi }_{lk},k=1,\dots ,a_l^d,l=1,2,\dots .$$

Here, $s>d-1$ is required to ensure that the trace of the correlation operator $C_v$ is finite. In other words, this condition guarantees that the Gaussian measure assigned is well defined. In fact, we have

$$\mathrm{Tr}\left(C_v\right)=\sum _{l=1}^{\infty }{a}_l^d{\zeta }_l^{-{\displaystyle \frac{s}{2}}}\asymp \sum _{l=1}^{\infty }{l}^{d-2}{\left(l(l+2{\lambda }_{\kappa })\right)}^{-{\displaystyle \frac{s}{2}}}\asymp \sum _{l=1}^{\infty }{l}^{d-s-2}.$$

Therefore, when $s>d-1$, the series converges.

It is well known that ${\langle C_{v}f,g\rangle }_r={\int }_{W_{2,\kappa }^r}{\langle f,h\rangle }_r{\langle g,h\rangle }_{r}v\left(\mathrm{d}h\right)$. Additionally, the Cameron–Martin space $H\left(v\right)$ associated with the Gaussian measure v is $W_{2,\kappa }^{r+{\displaystyle \frac{s}{2}}}$, as established in [19]; in other words, we have

$$H\left(v\right)=W_{2,\kappa }^{\rho }:=W_{2,\kappa }^{r+{\displaystyle \frac{s}{2}}}.$$

(4)

We then choose an any orthonormal system of functions $g_k,k=1,2,\dots ,n$ in $L_{2,\kappa }$ and let these $g_k$ belong to $H\left(v\right)$. Also, we let $\mathcal{D}$ be an any Borel subset of ${\mathbb{R}}^n$. Then, the Gaussian measure v of the cylindrical subset

$$G=\left\{f\in W_{2,\kappa }^r\mid \left(〈f,g_1^{\left(\rho \right)}〉,\dots ,〈f,g_n^{\left(\rho \right)}〉\right)\in \mathcal{D}\right\}$$

is equal to

$$v\left(G\right)={\left(2\pi \right)}^{-{\displaystyle \frac{n}{2}}}{\int }_{\mathcal{D}}exp\left(-{\displaystyle \frac{1}{2}}\sum _{i=1}^n{u}_i^2\right)\mathrm{d}{u}_1\cdots \mathrm{d}{u}_n.$$

For further details on Gaussian measures in the context of Banach spaces, please refer to [20,21].

This article aims to investigate the approximation problem in weighted Sobolev spaces $W_{2,\kappa }^r$ defined on ${\mathbb{S}}^{d-1}$. Now, we derive the following main results of this article.

Theorem 1. 

Let $r>(d-1+2s_{\kappa }){({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}_{+},s>d-1,\rho =r+{\displaystyle \frac{s}{2}}>{\displaystyle \frac{d-1}{2}}+(d-1)\mu {({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}_{+},\delta \in (0,{\displaystyle \frac{1}{2}}].$ Then, we have

$$Y_{n,\delta }(W_{2,\kappa }^r,v,L_{q,\kappa })\asymp \left\{\begin{array}{cc}{n}^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}\left(1+n^{-min\{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{q}}\}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}\right),\hfill & 1\le q<\infty ,\hfill \\ n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}\sqrt{ln{\displaystyle \frac{n}{\delta }}},\hfill & q=\infty ,\hfill \end{array}\right.$$

where $Y_{n,\delta }$ denotes ${\lambda }_{n,\delta }$ or $d_{\delta }^n$, ${\left(a\right)}_{+}:=max\{a,0\}$, $s_{\kappa }={\displaystyle \sum _{j=1}^d}{\kappa }_j-\underset{1\le j\le d}{min}{\kappa }_j$, and $\mu :=max\{2{\kappa }_i:i=1,\dots ,d\}$.

By employing the techniques from [7,22] and utilizing Theorem 1, we arrived at the following result. The proof details are omitted.

Theorem 2. 

Let $r>(d-1+2s_{\kappa }){({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}_{+},s>d-1,0<p<\infty ,\rho =r+{\displaystyle \frac{s}{2}}>{\displaystyle \frac{d-1}{2}}+(d-1)\mu {({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}_{+}.$ Then, we have

$$Y_n^{\left(a\right)}{(W_{2,\kappa }^r,v,L_{q,\kappa })}_p\asymp \left\{\begin{array}{cc}{n}^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}},\hfill & 1\le q<\infty ,\hfill \\ n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}\sqrt{ln\left(en\right)},\hfill & q=\infty ,\hfill \end{array}\right.$$

where $Y_n^{\left(a\right)}$ denotes ${\lambda }_n^{\left(a\right)}$ or $d_{\left(a\right)}^n$.

Remark 3. 

Regarding the results for the Kolmogorov width, when $1\le q\le 2$, using a similar approach to the one for proving the linear width in this paper, the following result can be obtained:

$$d_{n,\delta }(W_{2,\kappa }^r,v,L_{q,\kappa })\asymp n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}\left(1+n^{-{\displaystyle \frac{1}{2}}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}\right).$$

For the case where $2<q\le \infty$, the proof method for the upper bound presented in this paper does not apply. Therefore, this issue will be addressed in future work. We conjecture, however, that for $2<q\le \infty$, the Kolmogorov width result is as follows:

$$d_{n,\delta }(W_{2,\kappa }^r,v,L_{q,\kappa })\asymp n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}\left(1+n^{-{\displaystyle \frac{1}{2}}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}\right),$$

where the definitions of s, r, ρ, and δ are as described above.

3. Main Lemmas

This section presents some useful lemmas which play a vital role in proving the main theorems of this paper. We first introduce a nonnegative $C^{\infty }$ function $\eta$ on $[0,+\infty )$, which is supported on $[0,2]$, equal to one on $[0,1]$, and positive on $[0,2]$.

Let

$$L_{n,\eta }(x,y):=\sum _{j=0}^{\infty }\eta \left({\displaystyle \frac{j}{n}}\right)P_j(x,y),x,y\in {\mathbb{S}}^{d-1},$$

(5)

where $P_j(x,y):={\sum }_{k=1}^{a_j^d}{\varphi }_{jk}\left(x\right){\varphi }_{jk}\left(y\right)$. For functions $\Phi :{\mathbb{S}}^{d-1}\times {\mathbb{S}}^{d-1}\to \mathbb{R}$ and $f:{\mathbb{S}}^{d-1}\to \mathbb{R},$ we write

$$\Phi \ast f\left(x\right):={\int }_{{\mathbb{S}}^{d-1}}\Phi (x,y)f\left(y\right)W_{\kappa }\left(y\right)\mathrm{d}\sigma \left(y\right).$$

Clearly, for $P\in {\Pi }_n^d$, we have

$$P\left(x\right)=L_{n,\eta }\ast P\left(x\right),x\in {\mathbb{S}}^{d-1}.$$

(6)

For all $f\in L_{q,\kappa },1\le q\le \infty ,$ the following properties hold (see [23]):

(1)

For each $n\in \mathbb{N}$, we obtain the following result:

$$L_{n,\eta }\ast f\in {\Pi }_{2n-1}^d.$$

(2)

For each $n\in \mathbb{N}$, we obtain the following result:

$$\parallel L_{n,\eta }{ \ast f\parallel }_{q,\kappa }\ll {\parallel f\parallel }_{q,\kappa },\mathrm{and}\parallel f-L_{n,\eta }{ \ast f\parallel }_{q,\kappa }\ll \underset{g\in {\Pi }_n^d}{inf}{\parallel f-g\parallel }_{q,\kappa }.$$

(7)

In this paper, we still use the following classical metric on the sphere ${\mathbb{S}}^{d-1}$:

$$\rho (x,y):=arccos\left(\sum _{j=1}^d{x}_j{y}_j\right),x,y\in {\mathbb{S}}^{d-1}.$$

Let $B(x,t)=\{y\in {\mathbb{S}}^{d-1}:\rho (x,y)\le t\}$, where $x\in {\mathbb{S}}^{d-1}$, $t\in (0,{\displaystyle \frac{\pi }{2}}]$. According to [24], a weight function w defined on the unit sphere ${\mathbb{S}}^{d-1}$ is referred to as an $A_{\infty }$ weight if there exists a constant $\beta \ge 1$ such that for any spherical cap $C\subset {\mathbb{S}}^{d-1}$ and measurable set $F\subset C$, the following condition holds:

$${\displaystyle \frac{{\int }_{C}w\left(x\right)\mathrm{d}\sigma \left(x\right)}{{\int }_{F}w\left(x\right)\mathrm{d}\sigma \left(x\right)}}\le \beta {\left({\displaystyle \frac{{\int }_C\mathrm{d}\sigma \left(x\right)}{{\int }_F\mathrm{d}\sigma \left(x\right)}}\right)}^{\beta }.$$

From [24], it follows that $W_{\kappa }$ satisfies the above equation, and thus it is an $A_{\infty }$ weight.

Lemma 1 

([24]). For $1\le p,q\le \infty$, and $g\in {\Pi }_n^d$, $n\ge 1$. Then, we have

$${\parallel g\parallel }_{p,\kappa }\ll n^{(d-1+2s_{\kappa }){({\displaystyle \frac{1}{q}}-{\displaystyle \frac{1}{p}})}_{+}}{\parallel g\parallel }_{q,\kappa },$$

where $s_{\kappa }={\displaystyle \sum _{j=1}^d}{\kappa }_j-\underset{1\le j\le d}{min}{\kappa }_j$.

Lemma 2. 

Let $1\le q\le \infty ,r>(d-1+2s_{\kappa }){({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}_{+}.$ Then, $W_{2,\kappa }^r$ can be continuously embedded into the space $L_{q,\kappa }$.

Proof. 

We define

$$A_0\left(f\right)=L_{1,\eta } \ast f,A_j\left(f\right)=L_{2^j,\eta } \ast f-L_{2^{j-1},\eta } \ast f,\mathrm{for}j\ge 1.$$

Then, for any $f\in W_{2,\kappa }^r$, under Theorem 3.3 in [23] and Equation (7), we have

$$\parallel f-L_{n,\eta }{ \ast f\parallel }_{2,\kappa }\ll \underset{g\in {\Pi }_n^d}{inf}{\parallel f-g\parallel }_{2,\kappa }\ll n^{-r}\parallel f^{\left(r\right)}{\parallel }_{2,\kappa }=n^{-r}{\parallel f\parallel }_{W_{2,\kappa }^r}.$$

Hence, we obtain

$$\parallel A_j{\left(f\right)\parallel }_{2,\kappa }\le \parallel f-L_{2^j,\eta }{ \ast f\parallel }_{2,\kappa }+\parallel f-L_{2^{j-1},\eta }{ \ast f\parallel }_{2,\kappa }\ll 2^{-j}{\parallel f\parallel }_{W_{2,\kappa }^r}.$$

It can then be concluded through Lemma 1 that for $1\le q\le \infty$, we obtain

$$\parallel A_j{\left(f\right)\parallel }_{q,\kappa }\ll 2^{j(d-1+2s_{\kappa }){({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}_{+}}\parallel A_j{\left(f\right)\parallel }_{2,\kappa }\ll 2^{-jr+j(d-1+2s_{\kappa }){({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}_{+}}{\parallel f\parallel }_{W_{2,\kappa }^r}.$$

Therefore, for $r>(d-1+2s_{\kappa }){({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}_{+}$, we have

$${\parallel f\parallel }_{q,\kappa }\le \sum _{j=0}^{\infty }\parallel A_j{\left(f\right)\parallel }_{q,\kappa }\ll \sum _{j=0}^{\infty }{2}^{-jr+j(d+2s_{\kappa }){({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}_{+}}{\parallel f\parallel }_{W_{2,\kappa }^r}\le {\parallel f\parallel }_{W_{2,\kappa }^r}.$$

This concludes the proof. □

Lemma 3 

([2]). Let X be a normed linear space and $W_1$, $W_2$, and W be subsets of X. If $W_1\subset W_2\subset X$, then the following relations hold:

$$\begin{array}{cc}& {\lambda }_n(W_1,X)\le {\lambda }_n(W_2,X),d^n(W_1,X)=d^n(W_2,X),\hfill \\ & \mathit{and}{d}^n(W,X)\le {\lambda }_n(W,X).\hfill \end{array}$$

Lemma 4 

([2]). Let Y a normed linear space and X be a Hilbert space. If there exists a continuous embedding of X into Y, then the relation

$${\lambda }_n(X,Y)=d^n(X,Y)$$

holds true.

The following can be readily obtained by combining the proof of Theorem 1.5 in [7] with Lemmas 3 and 4. The proof is omitted here.

Lemma 5. 

Let Y be a normed linear space and X be a Hilbert space. If there exists a continuous embedding of X into Y, where $\delta \in (0,{\displaystyle \frac{1}{2}}]$, and v is a probabilistic measure on H, then the following result holds:

$${\lambda }_{n,\delta }(X,v,Y)=d_{\delta }^n(X,v,Y).$$

According to Lemma 5, to obtain Theorem 1, it is enough to consider the case of linear widths, which significantly reduces the amount of work required. The following will focus on the relevant lemmas concerning linear widths.

Consider a finite-dimensional ${\ell }_q$ space ${\ell }_q^m$ ($1\le q\le \infty$), which is the space ${\mathbb{R}}^m$ equipped with the ${\ell }_q^m$ norm, defined as follows:

$${\parallel x\parallel }_{{\ell }_q^m}:=\left\{\begin{array}{cc}{\left(\sum _{i=1}^m{\left|x_i\right|}^q\right)}^{{\displaystyle \frac{1}{q}}},\hfill & 1\le q<\infty ;\hfill \\ \underset{1\le i\le m}{max}\left|x_i\right|,\hfill & q=\infty .\hfill \end{array}\right.$$

Next, we review the standard Gaussian measure ${\gamma }_m$ in ${\mathbb{R}}^m$, which is defined by

$${\gamma }_m\left(G\right)={\left(2\pi \right)}^{-{\displaystyle \frac{m}{2}}}{\int }_{G}exp\left(-{\displaystyle \frac{{\parallel x\parallel }_{{\ell }_2^m}^2}{2}}\right)\mathrm{d}x,$$

where G is an arbitrary Borel subset in ${\mathbb{R}}^m$.

Let $1\le n<m$, $\delta \in [0,1)$, and $1\le q\le \infty$. Let $D=\mathrm{diag}(d_1,\dots ,d_m)$ be a diagonal operator of the order m, where $d_1\ge d_2\ge \cdots \ge d_m>0$. The linear $(n,\delta )$-width of D is defined as follows:

$${\lambda }_{n,\delta }(D:{\mathbb{R}}^m\to {\ell }_q^m,{\gamma }_m)=\underset{G_{\delta }}{inf}\underset{T_n}{inf}\underset{x\in {\mathbb{R}}^m\setminus G_{\delta }}{sup}{\parallel Dx-T_{n}y\parallel }_{{\ell }_q^m},$$

where $G_{\delta }$ denotes any Borel subset of ${\mathbb{R}}^m$ and satisfies ${\gamma }_m\left(G_{\delta }\right)\le \delta$ and $T_n$ represents linear operators from ${\mathbb{R}}^m$ to ${\ell }_q^m$ having a rank $T_n\le n$.

Lemma 6 

([5,8,22]). The following results hold:

(1)

Let $1\le q<\infty ,$ $\delta \in (0,{\displaystyle \frac{1}{2}}]$, and $m\ge 2n$. Then, the following result holds:

$${\lambda }_{n,\delta }(I_m:{\mathbb{R}}^m\to {\ell }_q^m,{\gamma }_m)\asymp m^{{\displaystyle \frac{1}{q}}}+m^{{({\displaystyle \frac{1}{q}}-{\displaystyle \frac{1}{2}})}_{+}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}.$$

(2)

Let $q=\infty$, $\delta \in (0,{\displaystyle \frac{1}{2}}]$, and $m\ge 2n$. Then, the following result holds:

$${\lambda }_{n,\delta }(I_m:{\mathbb{R}}^m\to {\ell }_q^m,{\gamma }_m)\asymp \sqrt{lnm+ln{\displaystyle \frac{1}{\delta }}}.$$

Lemma 7 

([13]). It is given that

$$\sum _{i=1}^m{d}_i^{\beta }\le C(m,\beta )\mathrm{for}\mathrm{some}\beta >0.$$

Then, for $m\ge 2n,\delta \in (0,{\displaystyle \frac{1}{2}}],$ we have

$${\lambda }_{n,\delta }(D:{\mathbb{R}}^m\to {\ell }_q^m,{\gamma }_m)\ll {\left({\displaystyle \frac{C(m,\beta )}{n+1}}\right)}^{{\displaystyle \frac{1}{\beta }}}\left\{\begin{array}{cc}\sqrt{m^{{\displaystyle \frac{2}{q}}}+ln{\displaystyle \frac{1}{\delta }}},& 2\le q<\infty ,\\ \\ \sqrt{lnm+ln{\displaystyle \frac{1}{\delta }}},& q=\infty .\end{array}\right.$$

4. Discretization

This section focuses on deriving the discretization theorems which simplify the calculation of the probabilistic widths of a function class to that of a finite-dimensional set under the standard Gaussian measure.

For any $f\in L_{2,\kappa }$, we define

$${\delta }_k\left(f\right)=S_{2^k}\left(f\right)-S_{2^{k-1}}\left(f\right)\mathrm{for}k=2,3\dots ,\mathrm{and}{\delta }_1\left(f\right)=S_2\left(f\right),$$

(8)

where $S_n$ is given in Equation (3). We denote by

$$M_k(x,y)=\sum _{j=2^{k-1}+1}^{2^k}\sum _{i=1}^{a_j^d}{\varphi }_{ji}\left(x\right){\varphi }_{ji}\left(y\right)$$

(9)

the reproducing kernel of the Hilbert space $L_{2,\kappa }\cap \left({⨁}_{j=2^{k-1}+1}^{2^k}{\mathcal{H}}_j^d\right).$ Then, we have ${\delta }_k\left(f\right)\left(x\right)=\langle f,M_k(·,x)\rangle ,$ $x\in {\mathbb{S}}^{d-1}.$ Moreover, if $g\in {⨁}_{j=2^{k-1}+1}^{2^k}{\mathcal{H}}_j^d$, we also have $g\left(x\right)=\langle g,M_k(·,x)\rangle .$

The following lemma is crucial for establishing the upper bound result in Theorem 1. Before presenting the lemma, we first introduce some related concepts.

A finite subset $\Lambda \subset {\mathbb{S}}^{d-1}$ is defined as maximal $\left({\displaystyle \frac{\delta }{n}},\rho \right)$-separable if it satisfies ${\mathbb{S}}^{d-1}\subset {\displaystyle \bigcup _{y\in \Lambda }}\mathit{B}\left(y,{\displaystyle \frac{\delta }{n}}\right)$ and $\underset{y\ne y^{\prime }\in \Lambda }{min}\rho (y,y^{\prime })\ge {\displaystyle \frac{\delta }{n}}.$

Lemma 8 

([24,25]). Let w be a doubling weight on ${\mathbb{S}}^{d-1}$. There exists a constant $\gamma >0$, which depends solely on d and the doubling constant of w, such that for every $0<\delta <\gamma$ and any maximal $\left({\displaystyle \frac{\delta }{n}},\rho \right)$-separable subset $\Lambda \subset {\mathbb{S}}^{d-1}$, a sequence of positive numbers ${\lambda }_{\omega }\asymp {\int }_{B(\omega ,{\displaystyle \frac{\delta }{n}})}w\left(y\right)\mathrm{d}\sigma \left(y\right)$ for $\omega \in \Lambda$ exists such that for any $f\in {\Pi }_n^d$, we have

$${\int }_{{\mathbb{S}}^{d-1}}f\left(x\right)w\left(x\right)\mathrm{d}\sigma \left(x\right)=\sum _{\omega \in \Lambda }{\lambda }_{\omega }f\left(\omega \right)$$

and moreover, for $0<p<\infty ,$ we have

$${\left({\int }_{{\mathbb{S}}^{d-1}}{\left|f\left(x\right)\right|}^{p}w\left(x\right)\mathrm{d}\sigma \left(x\right)\right)}^{{\displaystyle \frac{1}{p}}}\asymp {\left(\sum _{\omega \in \Lambda }{\lambda }_{\omega }{\left|f\left(\omega \right)\right|}^p\right)}^{{\displaystyle \frac{1}{p}}},$$

and

$$\underset{x\in {\mathbb{S}}^{d-1}}{max}\left|f\left(x\right)\right|\asymp \underset{\omega \in \Lambda }{max}\left|f\left(\omega \right)\right|,$$

where the constants of equivalence are determined solely by the doubling constant of w, as well as the parameters p and d.

It is well known that $A_{\infty }$ weights are special doubling weights, and thus the weight $W_{\kappa }^r$ satisfies the above lemma. Additionally, the above formulas are referred to as the cubature formula and the Marcinkiewicz–Zygmund inequalities, respectively.

Lemma 9 

([12]). Let $\mu :=max\{2{\kappa }_i:i=1,\dots ,d\}.$ For $\beta \in \left[0,{\displaystyle \frac{1}{\mu }}\right)$, the following inequality holds:

$$\sum _{\omega \in \Lambda }{\lambda }_{\omega }^{-\beta }\le c{n}^{(d-1)(1+\beta )},$$

where the definitions of Λ and ${\lambda }_{\omega }$ can be found in Lemma 8.

Let $\gamma >0$ be the constant as shown in Lemma 8 and denote ${\Lambda }_k=\left\{{\xi }_1,\dots ,{\xi }_{u_k}\right\}$ as a maximal $\left({\displaystyle \frac{\gamma }{10·2^k}},\rho \right)$-separated subset of ${\mathbb{S}}^{d-1},$ where $k=1,2,\dots .$ According to [12], it can be concluded that $u_k\asymp {\Pi }_{2^k}^d\asymp 2^{k(d-1)}$. According to Lemma 8, for all $f\in {\Pi }_{2^k}^d$, the following cubature formula holds:

$${\int }_{{\mathbb{S}}^{d-1}}f\left(x\right)W_{\kappa }\left(x\right)\mathrm{d}\sigma \left(x\right)=\sum _{i=1}^{u_k}{\omega }_{i}f\left({\xi }_i\right),$$

(10)

with ${\omega }_i\asymp {\int }_{B({\xi }_i,{\displaystyle \frac{\gamma }{10·2^k}})}W\left(y\right)\mathrm{d}\sigma \left(y\right)$. Additionally, for any $1\le q\le \infty$, it holds that

$${\parallel f\parallel }_{q,\kappa }\asymp {\left(\sum _{i=1}^{u_k}{\left|f\left({\xi }_i\right)\right|}^q{\omega }_i\right)}^{{\displaystyle \frac{1}{q}}}={∥U_k\left(f\right)∥}_{{\ell }_{q,\omega }^{u_k}}.$$

Here, $U_k:{\Pi }_{2^k}^d⟼{\mathbb{R}}^{u_k}$ is given as

$$U_k\left(f\right)=\left(f\left({\xi }_1\right),\dots ,f\left({\xi }_{u_k}\right)\right),$$

and for any $y\in {\mathbb{R}}^{u_k}$, we have

$${\parallel y\parallel }_{{\ell }_{q,\omega }^{u_k}}:=\left\{\begin{array}{cc}{\left(\sum _{i=1}^{u_k}{\left|y_i\right|}^q{\omega }_i\right)}^{{\displaystyle \frac{1}{q}}},\hfill & 1\le q<\infty ;\hfill \\ \underset{1\le i\le u_k}{max}\left|y_i\right|,\hfill & q=\infty .\hfill \end{array}\right.$$

Obviously, $U_k$ is a linear operator. Now, we introduce the linear operator $T_k:{\mathbb{R}}^{u_k}⟼{\Pi }_{2^{k+1}}^d$ as

$$T_{k}a\left(x\right):=\sum _{i=1}^{u_k}{a}_i{\omega }_i{L}_{2^k,\eta }\left(x,{\xi }_i\right).$$

(11)

Here, $a=\left(a_1,\dots ,a_{u_k}\right)\in {\mathbb{R}}^{u_k},$ and $L_{2^k,\eta }$ is given as shown in Equation (5). It is demonstrated in Equation (5.7) of [12] that for any $1\le q\le \infty ,$ one has

$${∥T_{k}a∥}_{q,\kappa }\ll {\parallel a\parallel }_{{\ell }_{q,\omega }^{u_k}}.$$

(12)

It follows directly from Equations (6), (10), and (11) that $f=T_k{U}_{k}f$ for any $f\in {\Pi }_{2^k}^d$.

One of the main theorems in this section is presented below, which is essential for estimating the upper bound in Theorem 1 and is referred to as the upper bound estimation discretization theorem.

Theorem 3. 

Let $1\le q\le \infty ,\sigma \in (0,{\displaystyle \frac{1}{2}}]$, along with sequences $\left\{n_k\right\}$ and $\left\{{\sigma }_k\right\}$ satisfying the conditions $0\le n_k\le u_k\asymp 2^{k(d-1)},{\sigma }_k\in (0,{\displaystyle \frac{1}{2}}],{\sum }_{k=1}^{\infty }{n}_k\le n,$ and ${\sum }_{k=1}^{\infty }{\sigma }_k\le \sigma .$ Then, we have

$${\lambda }_{n,\sigma }(W_{2,\kappa }^r,v,L_{q,k})\ll \sum _{k=1}^{\infty }{2}^{-k\rho }{\lambda }_{n_k,{\sigma }_k}(V_k:{\mathbb{R}}^{u_k}\to {\ell }_q^{u_k},{\gamma }_{u_k}),$$

(13)

where $V_k=diag\left({\omega }_1^{-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{q}}},\dots ,{\omega }_{u_k}^{-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{q}}}\right)$.

The following lemma will be crucial in assisting with the proof of Theorem 3.

Lemma 10. 

For any $z=(z_1,\dots ,z_{u_k})\in {\mathbb{R}}^{u_k}$, we have

$${∥\sum _{j=1}^{u_k}{\omega }_j^{{\displaystyle \frac{1}{2}}}{z}_j{M}_k(·,{\xi }_j)∥}_{2,k}\ll {∥z∥}_{{\ell }_2^{u_k}},$$

where $\{{\xi }_1,\dots ,{\xi }_{u_k}\}$ is as previously defined and $M_k(x,y)$ is defined by Equation (9).

Proof. 

Let K represent the set

$$\{g\in \underset{j=2^{k-1}+1-2{\delta }_{k,1}}{\overset{2^k}{⨁}}{\mathcal{H}}_j^d\mid \parallel g{\parallel }_{2,k}\le 1\}.$$

We know that

$$\sum _{j=1}^{u_k}{\omega }_j^{{\displaystyle \frac{1}{2}}}{z}_j{M}_k(·,{\xi }_j)\in \underset{j=2^{k-1}+1-2{\delta }_{k,1}}{\overset{2^k}{⨁}}{\mathcal{H}}_j^d,$$

and thus, through the Riesz representation theorem, the expression in Equation (10), and the Cauchy–Schwartz inequality, we obtain

$$\begin{array}{c}{∥{\displaystyle \sum _{j=1}^{u_k}}{\omega }_j^{{\displaystyle \frac{1}{2}}}{z}_j{M}_n(·,{\xi }_j)∥}_{2,k}\begin{array}{c}\hfill =\underset{g\in K}{sup}\left|\langle \sum _{j=1}^{u_k}{\omega }_j^{{\displaystyle \frac{1}{2}}}{z}_j{M}_k(·,{\xi }_j),g\rangle \right|={\displaystyle \underset{g\in K}{sup}}\left|\sum _{j=1}^{u_k}{\omega }_j^{{\displaystyle \frac{1}{2}}}{z}_{j}g\left({\xi }_j\right)\right|\end{array}\\ \le {\displaystyle \underset{g\in K}{sup}}{\left({\displaystyle \sum _{j=1}^{u_k}}{\left|z_j\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}{\left({\displaystyle \sum _{j=1}^{u_k}}{\left|g\left({\xi }_j\right)\right|}^2{\omega }_j\right)}^{{\displaystyle \frac{1}{2}}}\\ \ll {\displaystyle \underset{g\in K}{sup}}{\left({\displaystyle \sum _{j=1}^{u_k}}{\left|z_j\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}{\parallel g\parallel }_{2,k}\le {\parallel z\parallel }_{{\ell }_2^{u_k}}.\end{array}$$

This concludes the proof. □

We are now prepared to begin the proof of Theorem 3.

Proof of Theorem 3. 

First, we write

$${\lambda }_{n_k,{\sigma }_k}:={\lambda }_{n_k,{\sigma }_k}(V_k:{\mathbb{R}}^{u_k}\to {\ell }_q^{u_k},{\gamma }_{u_k}),$$

where ${\gamma }_{u_k}$ is the standard Gaussian measure in ${\mathbb{R}}^{u_k}$. Let $L_k$ be a linear operator from ${\mathbb{R}}^{u_k}$ to ${\mathbb{R}}^{u_k}$ with a rank $L_k\le n_k$ and

$${\gamma }_{u_k}\left(\{y\in {\mathbb{R}}^{u_k}\mid \parallel V_{k}y-L_{k}y{\parallel }_{{\ell }_q^{u_k}}>2{\lambda }_{n_k,{\sigma }_k}\}\right)\le {\sigma }_k.$$

Recalling the definition of ${\delta }_k$ (see Equation (8)), it follows that ${\delta }_k\left(f\right)\in {\Pi }_{2^k}^d$ for $f\in W_{2,\kappa }^r$. Therefore, $T_k{U}_k{\delta }_k\left(f\right)={\delta }_k\left(f\right)$.

We denote

$$y:=S_k{U}_k{\delta }_k\left(f\right)=({\omega }_1^{{\displaystyle \frac{1}{2}}}{\delta }_k\left(f\right)\left({\xi }_1\right),\dots ,{\omega }_{u_k}^{{\displaystyle \frac{1}{2}}}{\delta }_k\left(f\right)\left({\xi }_{u_k}\right))\in {\mathbb{R}}^{u_k},$$

where $S_k=diag\left({\omega }_1^{{\displaystyle \frac{1}{2}}},\dots ,{\omega }_{u_k}^{{\displaystyle \frac{1}{2}}}\right).$ We also have the following equation:

$$\begin{array}{cc}\hfill \parallel {\delta }_k\left(f\right)-T_k{R}_k^{-1}{L}_k{y\parallel }_{q,\kappa }& =\parallel T_k{U}_k{\delta }_k\left(f\right)-T_k{R}_k^{-1}{L}_k{y\parallel }_{q,\kappa }\hfill \\ \hfill & \ll \parallel U_k{\delta }_k\left(f\right)-R_k^{-1}{L}_k{y\parallel }_{{\ell }_{q,\omega }^{u_k}}\hfill \\ \hfill & ={∥V_{k}y-L_{k}y∥}_{{\ell }_q^{u_k}},\hfill \end{array}$$

(14)

Here, $R_k=diag\left({\omega }_1^{{\displaystyle \frac{1}{q}}},\dots ,{\omega }_{u_k}^{{\displaystyle \frac{1}{q}}}\right)$, and the second inequality follows from Equation (12). For $x\in {\mathbb{S}}^{d-1}$, we have

$$\begin{array}{cc}\hfill {\delta }_k\left(f\right)\left(x\right)& =\langle f,M_k(·,x)\rangle ={\langle f^{(-r)},M_k^{(-r,0)}(·,x)\rangle }_r\hfill \\ & ={\langle f,M_k^{(-2r,0)}(·,x)\rangle }_r,\hfill \end{array}$$

where $r\in \mathbb{R}$ and $M_k^{(r,0)}(x,y)$ is the rth partial derivative of $M_k(x,y)$ with respect to the first variable. Since the random vector $f\in W_{2,\kappa }^r$ is a centered Gaussian random vector with a covariance operator $C_v$, the random vector $y=S_k{U}_k{\delta }_k\left(f\right)$, obtained through a linear transformation, remains a centered Gaussian random vector. Its covariance matrix is denoted by $C_{\gamma }$. Note that

$$y=({\langle f,{\omega }_1^{{\displaystyle \frac{1}{2}}}{M}_k^{(-2r,0)}(·,{\xi }_1)\rangle }_r,\dots ,{\langle f,{\omega }_{u_k}^{{\displaystyle \frac{1}{2}}}{M}_k^{(-2r,0)}(·,{\xi }_{u_k})\rangle }_r)\in {\mathbb{R}}^{u_k}.$$

Hence, its covariance matrix $C_{\gamma }$ is expressed as follows:

$$C_{\gamma }={\left({\langle C_v\left({\omega }_i^{{\displaystyle \frac{1}{2}}}{M}_k^{(-2r,0)}(·,{\xi }_i)\right),{\omega }_j^{{\displaystyle \frac{1}{2}}}{M}_k^{(-2r,0)}(·,{\xi }_j)\rangle }_r\right)}_{i,j=1}^{u_k}.$$

It is quite straightforward to observe that for any $z=(z_1,\dots ,z_{u_k})\in {\mathbb{R}}^{u_k},$ we have

$$\sum _{j=1}^{u_k}{\omega }_j^{{\displaystyle \frac{1}{2}}}{z}_j{M}_k(·,{\xi }_j)\in \underset{j=2^{k-1}+1-2{\delta }_{k,1}}{\overset{2^k}{⨁}}{\mathcal{H}}_j^d,$$

and

$$\begin{array}{c}{〈C_v\left({\omega }_i^{{\displaystyle \frac{1}{2}}}{M}_k^{(-2r,0)}(·,{\xi }_i)\right),{\omega }_j^{{\displaystyle \frac{1}{2}}}{M}_k^{(-2r,0)}(·,{\xi }_j)〉}_r\\ ={〈{\omega }_i^{{\displaystyle \frac{1}{2}}}{M}_k^{(-2r-s,0)}(·,{\xi }_i),{\omega }_j^{{\displaystyle \frac{1}{2}}}{M}_k^{(-2r,0)}(·,{\xi }_j)〉}_r\\ =〈{\omega }_i^{{\displaystyle \frac{1}{2}}}{M}_k^{(-\rho ,0)}(·,{\xi }_i),{\omega }_j^{{\displaystyle \frac{1}{2}}}{M}_k^{(-\rho ,0)}(·,{\xi }_j)〉.\end{array}$$

Under Lemma 10, we have

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^{u_k}}{(y,z)}^2\gamma \left(\mathrm{d}y\right)& =z{C}_{\gamma }{z}^T=\sum _{i,j=1}^{u_k}{z}_i{z}_j〈{\omega }_i^{{\displaystyle \frac{1}{2}}}{M}_k^{(-\rho ,0)}(·,{\xi }_i),{\omega }_j^{{\displaystyle \frac{1}{2}}}{M}_k^{(-\rho ,0)}(·,{\xi }_j)〉\hfill \\ \hfill & =〈\sum _{j=1}^{u_k}{\omega }_j^{{\displaystyle \frac{1}{2}}}{z}_j{M}_k^{(-\rho ,0)}(·,{\xi }_j),\sum _{j=1}^{u_k}{\omega }_j^{{\displaystyle \frac{1}{2}}}{z}_j{M}_k^{(-\rho ,0)}(·,{\xi }_j)〉\hfill \\ \hfill & ={∥\sum _{j=1}^{u_k}{\omega }_j^{{\displaystyle \frac{1}{2}}}{z}_j{M}_k^{(-\rho ,0)}(·,{\xi }_j)∥}_{2,\kappa }^2\hfill \\ \hfill & \asymp 2^{-2k\rho }{∥\sum _{j=1}^{u_k}{\omega }_j^{{\displaystyle \frac{1}{2}}}{z}_j{M}_k(·,{\xi }_j)∥}_{2,\kappa }^2\hfill \\ \hfill & \ll 2^{-2k\rho }{\parallel z\parallel }_{{\ell }_2^{u_k}}^2=2^{-2k\rho }{\int }_{{\mathbb{R}}^{u_k}}{(y,z)}^2{\gamma }_{u_k}\left(\mathrm{d}y\right).\hfill \end{array}$$

(15)

The subset $G_k$ of $W_{2,\kappa }^r$ is defined as follows:

$$G_k:=\{f\in W_{2,\kappa }^r\mid \parallel {\delta }_k\left(f\right)-T_k{R}_k^{-1}{L}_k{S}_k{U}_k{\delta }_k\left(f\right){\parallel }_{q,\kappa }>2c_1{c}_2{2}^{-k\rho }{\lambda }_{n_k,{\sigma }_k}\},$$

where $c_1$ and $c_2$ are the positive constants given in Equations (14) and (15). From Equation (14), we obtain

$$\begin{array}{cc}\hfill v\left(G_k\right)& \le v\left(\left\{f\in W_{2,\kappa }^r\mid {\parallel V_k{S}_k{U}_k{\delta }_k\left(f\right)-L_k{S}_k{U}_k{\delta }_k\left(f\right)\parallel }_{{\ell }_q^{u_k}}>2c_2{2}^{-k\rho }{\lambda }_{n_k,{\sigma }_k}\right\}\right)\hfill \\ & =\gamma \left(\left\{y\in {\mathbb{R}}^{u_k}\mid {\parallel V_{k}y-L_{k}y\parallel }_{{\ell }_q^{u_k}}>2c_2{2}^{-k\rho }{\lambda }_{n_k,{\sigma }_k}\right\}\right).\hfill \end{array}$$

It is easy to see that the set $\{y\in {\mathbb{R}}^{u_k}\mid \parallel V_{k}y-L_{k}y{\parallel }_{{\ell }_q^{u_k}}\le t\}$ is both convex and symmetric for any $t>0$. Moreover, since Equation (15) holds, according to the Gaussian measure comparison theorem (see [19], p. 29), the following equation holds:

$$\begin{array}{cc}\hfill v\left(G_k\right)& \le \gamma \left(\left\{y\in {\mathbb{R}}^{u_k}\mid {\parallel V_{k}y-L_{k}y\parallel }_{{\ell }_q^{u_k}}>2c_2{2}^{-k\rho }{\lambda }_{n_k,{\sigma }_k}\right\}\right)\hfill \\ & \le \lambda \left(\left\{y\in {\mathbb{R}}^{u_k}\mid {\parallel V_{k}y-L_{k}y\parallel }_{{\ell }_q^{u_k}}>2c_2{2}^{-k\rho }{\lambda }_{n_k,{\sigma }_k}\right\}\right)\hfill \\ & ={\gamma }_{u_k}\left(\left\{y\in {\mathbb{R}}^{u_k}\mid {\parallel V_{k}y-L_{k}y\parallel }_{{\ell }_q^{u_k}}>2{\lambda }_{n_k,{\sigma }_k}\right\}\right)\hfill \\ & \le {\sigma }_k,\hfill \end{array}$$

where $\lambda$ is a centered Gaussian measure in ${\mathbb{R}}^{u_k}$ with a covariance matrix $c_2^2{2}^{-2k\rho }{I}_{u_k}$. We define $G={\bigcup }_{k=1}^{\infty }{G}_k$, and the linear operator $T_n$ on $W_{2,\kappa }^r$ is expressed by

$$T_{n}f=\sum _{k=1}^{\infty }{T}_k{R}_k^{-1}{L}_k{S}_k{U}_k{\delta }_k\left(f\right).$$

Hence, we have

$$v\left(G\right)=v\left(\bigcup _{k=1}^{\infty }{G}_k\right)\le \sum _{k=1}^{\infty }{\sigma }_k\le \sigma ,$$

and

$$rank{T}_n\le \sum _{k=1}^{\infty }rank\left(T_k{R}_k^{-1}{L}_k{S}_k{U}_k{\delta }_k\right)\le \sum _{k=1}^{\infty }{n}_k\le n.$$

Finally, according to the definitions of $G,T_n,\left\{G_k\right\},\mathrm{and}\left\{L_k\right\},$ we obtain

$$\begin{array}{cc}\hfill {\lambda }_{n,\delta }\left(W_{2,\kappa }^r,v,L_{q,\kappa }\right)& \le \underset{f\in W_{2,\kappa }^r\setminus G}{sup}{\parallel f-T_{n}f\parallel }_{q,\kappa }\hfill \\ & \le \underset{f\in W_{2,\kappa }^r\setminus G}{sup}\sum _{k=1}^{\infty }{∥{\delta }_k\left(f\right)-T_k{R}_k^{-1}{L}_k{S}_k{U}_k{\delta }_k\left(f\right)∥}_{q,\kappa }\hfill \\ & \le \sum _{k=1}^{\infty }\underset{f\in W_{2,\kappa }^r\setminus G_k}{sup}{∥{\delta }_k\left(f\right)-T_k{R}_k^{-1}{L}_k{S}_k{U}_k{\delta }_k\left(f\right)∥}_{q,\kappa }\hfill \\ & \ll \sum _{k=1}^{\infty }{2}^{-k\rho }{\lambda }_{n_k,{\sigma }_k}.\hfill \end{array}$$

This concludes the proof. □

We now turn to the lower bound estimation. Let m be large enough and $b_1{m}^{d-1}\le n\le b_2{m}^{d-1}$, where $b_1$, $b_2>0$ are constants which do not depend on n or m. Let ${\left\{x_j\right\}}_{j=1}^N\subset {\mathbb{S}}^{d-1}$, with $N\asymp m^{d-1}$, such that

$$B(x_i,{\displaystyle \frac{2}{m}})\cap B(x_j,{\displaystyle \frac{2}{m}})=\varnothing \mathrm{for}i\ne j.$$

We choose $b_2$ to be small enough to ensure that $N\ge 2n$. Let ${\phi }^1$ be a smooth function defined on $\mathbb{R}$ which is supported on the interval $[0,1]$ and takes the value of one on the subinterval $[0,{\displaystyle \frac{2}{3}}]$. Additionally, let ${\phi }^2$ be a nonnegative smooth function on $\mathbb{R}$, which is supported on the interval $[0,{\displaystyle \frac{1}{2}}]$ and equals one on the smaller subinterval $[0,{\displaystyle \frac{1}{4}}]$.

We then consider

$${\phi }_i\left(x\right)={\phi }^1(m·\rho (x,x_i))-c_i{\phi }^2(m·\rho (x,x_i)),$$

where $c_i$ is a constant chosen to satisfy

$${\int }_{{\mathbb{S}}^{d-1}}{\phi }_i\left(x\right)W_{\kappa }\left(x\right)\mathrm{d}\sigma \left(x\right)=0,i=1,\dots ,N.$$

Then, we let

$${\mathcal{A}}_N:=\left\{f_a\left(x\right)=\sum _{j=1}^N{a}_j{\phi }_j\left(x\right):a=(a_1,\dots ,a_N)\right\}.$$

It is clear that

$$supp{\phi }_i\subset B(x_i,{\displaystyle \frac{1}{m}}),$$

and

$$\parallel {\phi }_i{\parallel }_{p,\kappa }\asymp {\left({\int }_{B(x_i,{\displaystyle \frac{1}{m}})}{\left|{\phi }^1(m·\rho (x,x_i))-c_i{\phi }^2(m·\rho (x,x_i))\right|}^p\mathrm{d}\sigma \left(x\right)\right)}^{{\displaystyle \frac{1}{p}}}\asymp m^{-{\displaystyle \frac{d-1}{p}}},$$

with

$$supp{\phi }_i\cap supp{\phi }_j=\varnothing \mathrm{if}i\ne j.$$

For $f_a\in {\mathcal{A}}_N$ with $a=(a_1,\dots ,a_N)$, we have

$$\parallel f_a{\parallel }_{p,\kappa }\asymp {\left(m^{-(d-1)}\sum _{j=1}^N{\left|a_j\right|}^p\right)}^{{\displaystyle \frac{1}{p}}}=m^{-{\displaystyle \frac{d-1}{p}}}{\parallel a\parallel }_{{\ell }_p^N}.$$

(16)

From [12], it is known that for a positive integer $v>{\displaystyle \frac{r}{2}}$, there exists

$$\parallel {(-D_{\kappa })}^v\left(f_a\right){\parallel }_{p,\kappa }\ll m^{-{\displaystyle \frac{d-1}{p}}+2v}{\parallel a\parallel }_{{\ell }_p^N}.$$

Therefore, under the Kolmogorov-type inequality (see [26]), we obtain

$$\begin{array}{cc}\hfill \parallel f_a^{\left(\rho \right)}{\parallel }_{p,\kappa }& =\parallel {(-D_{\kappa }^d)}^{{\displaystyle \frac{\rho }{2}}}\left(f_a\right){\parallel }_{p,\kappa }\hfill \\ \hfill & \ll \parallel {(-D_{\kappa }^d)}^{1+\left[\rho \right]}\left(f_a\right){\parallel }_{p,\kappa }^{{\displaystyle \frac{\rho }{2+2\left[\rho \right]}}}{\parallel f_a\parallel }_{p,\kappa }^{1-{\displaystyle \frac{\rho }{2+2\left[\rho \right]}}}\hfill \\ \hfill & \ll m^{\rho -{\displaystyle \frac{d-1}{p}}}{\parallel a\parallel }_{{\ell }_p^N}\ll m^{\rho }{\parallel f_a\parallel }_{p,\kappa },\hfill \end{array}$$

(17)

where $\left[a\right]$ denotes the greatest integer b such that $b\le a$.

For $f\in L_{1,\kappa }$ and $x\in {\mathbb{S}}^{d-1}$, we let

$$P_N\left(f\right)\left(x\right)=\sum _{j=1}^N{\displaystyle \frac{{\phi }_j\left(x\right)}{\parallel {\phi }_j{\parallel }_{2,\kappa }^2}}{\int }_{{\mathbb{S}}^{d-1}}f\left(y\right){\phi }_j\left(y\right)W_{\kappa }\left(y\right)\mathrm{d}\sigma \left(y\right),$$

and

$$Q_N\left(f\right)\left(x\right)=\sum _{j=1}^N{\displaystyle \frac{{\phi }_j\left(x\right)}{\parallel {\phi }_j{\parallel }_{2,\kappa }^2}}{\int }_{{\mathbb{S}}^{d-1}}f\left(y\right){\phi }_j^{\left(\rho \right)}\left(y\right)W_{\kappa }\left(y\right)\mathrm{d}\sigma \left(y\right).$$

The operator $P_N$ serves as an orthogonal projection from $L_{2,\kappa }$ onto ${\mathcal{A}}_N$. For $f\in W_{2,\kappa }^{\rho }$, we have $Q_N\left(f\right)\left(x\right)=P_N\left(f^{\left(\rho \right)}\right)\left(x\right)$. Additionally, the following conclusion holds:

$$\parallel P_N{f\parallel }_{q,\kappa }\ll {\parallel f\parallel }_{q,\kappa },1\le q\le \infty .$$

(18)

This result can be derived as follows. The operator $P_N$ is given by

$$P_N\left(f\right)\left(x\right)={\int }_{{\mathbb{S}}^{d-1}}f\left(y\right)\mathcal{K}(x,y)W_{\kappa }\left(y\right)\mathrm{d}\sigma \left(y\right),$$

where $\mathcal{K}(x,y)={\sum }_{j=1}^N{\displaystyle \frac{{\phi }_j\left(x\right){\phi }_j\left(y\right)}{\parallel {\phi }_j{\parallel }_{2,\kappa }^2}}$. Using Equation (16), we find that

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{S}}^{d-1}}\left|\mathcal{K}(x,y)\right|W_{\kappa }\left(y\right)\mathrm{d}\sigma \left(y\right)& \ll m^{-(d-1)}\sum _{j=1}^N{\displaystyle \frac{|{\phi }_j\left(x\right)|}{\parallel {\phi }_j{\parallel }_{2,\kappa }^2}}\hfill \\ & \ll \sum _{j=1}^N\left|{\phi }_j\left(x\right)\right|\ll 1,\forall x\in {\mathbb{S}}^{d-1},\hfill \end{array}$$

where the last inequality holds because $supp{\phi }_i\cap supp{\phi }_j=\varnothing$ when $i\ne j$.

With the Hölder’s inequality, we estimate

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{S}}^{d-1}}{\left|P_N\left(f\right)\left(x\right)\right|}^q{W}_{\kappa }\left(x\right)\mathrm{d}\sigma \left(x\right)& ={\int }_{{\mathbb{S}}^{d-1}}{\left|{\int }_{{\mathbb{S}}^{d-1}}f\left(y\right)\mathcal{K}(x,y)W_{\kappa }\left(y\right)\mathrm{d}\sigma \left(y\right)\right|}^q{W}_{\kappa }\left(x\right)\mathrm{d}\sigma \left(x\right)\hfill \\ & \le {\int }_{{\mathbb{S}}^{d-1}}\left({\int }_{{\mathbb{S}}^{d-1}}{\left|f\left(y\right)\right|}^q\left|\mathcal{K}(x,y)\right|W_{\kappa }\left(y\right)\mathrm{d}\sigma \left(y\right)\right)\hfill \\ & ·{\left({\int }_{{\mathbb{S}}^{d-1}}\left|\mathcal{K}(x,y)\right|W_{\kappa }\left(y\right)\mathrm{d}\sigma \left(y\right)\right)}^{{\displaystyle \frac{q}{q^{\prime }}}}{W}_{\kappa }\left(x\right)\mathrm{d}\sigma \left(x\right)\hfill \\ & \ll {\int }_{{\mathbb{S}}^{d-1}}\left({\int }_{{\mathbb{S}}^{d-1}}{\left|f\left(y\right)\right|}^q\left|\mathcal{K}(x,y)\right|W_{\kappa }\left(y\right)\mathrm{d}\sigma \left(y\right)\right)W_{\kappa }\left(x\right)\mathrm{d}\sigma \left(x\right)\hfill \\ & ={\int }_{{\mathbb{S}}^{d-1}}\left({\int }_{{\mathbb{S}}^{d-1}}\left|\mathcal{K}(x,y)\right|W_{\kappa }\left(x\right)\mathrm{d}\sigma \left(x\right)\right){\left|f\left(y\right)\right|}^q{W}_{\kappa }\left(y\right)\mathrm{d}\sigma \left(y\right)\hfill \\ & \ll {\int }_{{\mathbb{S}}^{d-1}}{\left|f\left(y\right)\right|}^q{W}_{\kappa }\left(y\right)\mathrm{d}\sigma \left(y\right),\hfill \end{array}$$

where ${\displaystyle \frac{1}{q}}+{\displaystyle \frac{1}{q^{\prime }}}=1$ and $1\le q\le \infty$. Thus, the inequality in Equation (18) is verified.

If $f\in W_{2,\kappa }^{\rho }$, then it is clear that $Q_N\left(f\right)\in {\mathcal{A}}_N$. Using Equation (17), the following inequality holds:

$$\begin{array}{cc}\hfill \parallel {\left(Q_N\left(f\right)\right)}^{\left(\rho \right)}{\parallel }_{2,\kappa }& \ll m^{\rho }\parallel Q_N{\left(f\right)\parallel }_{2,\kappa }=m^{\rho }{\parallel P_N\left(f^{\left(\rho \right)}\right)\parallel }_{2,\kappa }\hfill \\ \hfill & \ll m^{\rho }{\parallel f^{\left(\rho \right)}\parallel }_{2,\kappa }.\hfill \end{array}$$

(19)

Another main theorem in this section is presented below, which is vital for estimating the lower bound in Theorem 1 and is referred to as the lower bound estimation discretization theorem.

Theorem 4. 

Let $1\le q\le \infty$, $\delta \in (0,{\displaystyle \frac{1}{2}}]$ and N be defined as described previously, satisfying $N\asymp n$ and $N\ge 2n$. Then, we have

$${\lambda }_{n,\delta }(W_{2,\kappa }^r,v,L_{q,\kappa })\gg n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}}}{\lambda }_{n,\delta }(I_N:{\mathbb{R}}^N\to {\ell }_q^N,{\gamma }_N),$$

where ${\gamma }_N$ denotes the standard Gaussian distribution in ${\mathbb{R}}^N$.

Proof. 

For convenience, we write

$${\lambda }_{n,\delta }:={\lambda }_{n,\delta }(W_{2,\kappa }^r,v,L_{q,\kappa }).$$

We denote $T_n$ as a bounded linear operator acting on $W_{2,\kappa }^r$, and the rank of $T_n$ does not exceed n such that

$$v\left(\left\{f\in W_{2,\kappa }^r:{\parallel f-T_{n}f\parallel }_{q,\kappa }>2{\lambda }_{n,\delta }\right\}\right)\le \delta .$$

If A is a bounded linear operator acting on $W_{2,\kappa }^r$ and $H\left(v\right)$, then the image measure $\lambda$ of v under A remains a centered Gaussian measure on $W_{2,\kappa }^r$, and its covariance takes the following form:

$$R_{\lambda }\left(f\right)\left(f\right)={\langle A^{\ast }{C}_{v}f,A^{\ast }{C}_{v}f\rangle }_{H\left(v\right)},f\in {\left(W_{2,\kappa }^r\right)}^{\ast }=W_{2,\kappa }^r,$$

where $A^{\ast }$ is the adjoint operator in $H\left(v\right)$, $C_v$ is the covariance of v, and $H\left(v\right)$ is given in Equation (4) (see [19], Theorem 3.5.1). Additionally, if A satisfies

$${\parallel Af\parallel }_{H\left(v\right)}\le {\parallel f\parallel }_{H\left(v\right)},$$

then

$$R_{\lambda }\left(f\right)\left(f\right)=\parallel A^{\ast }{C}_v{f\parallel }_{H\left(v\right)}^2\le \parallel A^{\ast }{\parallel }^2{\parallel C_{v}f\parallel }_{H\left(v\right)}\le {\langle C_{v}f,C_{v}f\rangle }_{H\left(v\right)}=R_v\left(f\right)\left(f\right).$$

According to [19] (Theorem 3.3.6), if for any Borel set $E\subset W_{2,\kappa }^r$ it is absolutely convex, then the following inequality holds:

$$v\left(E\right)\le \lambda \left(E\right).$$

From Equation (19), we have

$$\parallel Q_N{\left(f\right)\parallel }_{H\left(v\right)}=\parallel {\left(Q_N\left(f\right)\right)}^{\left(\rho \right)}{\parallel }_{2,\kappa }\ll m^{\rho }\parallel f^{\left(\rho \right)}{\parallel }_{2,\kappa }=m^{\rho }{\parallel f\parallel }_{H\left(v\right)}.$$

Thus, the constant $c_3>0$ exists such that

$${∥{\displaystyle \frac{1}{c_3{m}^{\rho }}}{Q}_N\left(f\right)∥}_{H\left(v\right)}\le {\parallel f\parallel }_{H\left(v\right)}.$$

Next, the set $\{f\in W_{2,\kappa }^r:\parallel f-T_{n}f{\parallel }_{q,\kappa }\le t\}$ is absolutely convex for $t>0$, and therefore

$$\begin{array}{cc}\hfill v\left(\right\{f& \in W_{2,\kappa }^r:{\parallel f-T_{n}f\parallel }_{q,\kappa }>2{\lambda }_{n,\delta }\left\}\right)\hfill \\ & \ge \lambda \left(\{f\in W_{2,\kappa }^r:\parallel f-T_{n}f{\parallel }_{q,\kappa }>2{\lambda }_{n,\delta }\right\})\hfill \\ & =v\left(\{f\in W_{2,\kappa }^r:\parallel Q_{N}f-T_n{Q}_{N}f{\parallel }_{q,\kappa }>2c_3{m}^{\rho }{\lambda }_{n,\delta }\right)).\hfill \end{array}$$

Let $J_N:{\mathcal{A}}_N\to {\mathbb{R}}^N$ and $L_N:{\mathbb{R}}^N\to {\mathcal{A}}_N$ be defined as follows:

$$J_N\left(f_a\right)=(a_1\parallel {\phi }_1{\parallel }_{2,\kappa },\dots ,a_N\parallel {\phi }_N{\parallel }_{2,\kappa }),f_a\in {\mathcal{A}}_N.$$

In addition, let

$$L_N\left(a\right)\left(x\right)=\sum _{i=1}^N{\displaystyle \frac{a_i{\phi }_i\left(x\right)}{\parallel {\phi }_i{\parallel }_{2,\kappa }}},a=(a_1,\dots ,a_N)\in {\mathbb{R}}^N,$$

It is clear that $L_N{J}_N\left(f_a\right)=f_a$ if $f_a\in {\mathcal{A}}_N$.

For $y=J_N{Q}_N\left(f\right)$, we have

$$y=\left({\displaystyle \frac{1}{\parallel {\phi }_1{\parallel }_{2,\kappa }}}\langle f,{\phi }_1^{\left(\rho \right)}\rangle ,\dots ,{\displaystyle \frac{1}{\parallel {\phi }_N{\parallel }_{2,\kappa }}}\langle f,{\phi }_N^{\left(\rho \right)}\rangle \right)\in {\mathbb{R}}^N.$$

Using Equation (16) and the fact that $\parallel {\phi }_j{\parallel }_{2,\kappa }\asymp m^{-{\displaystyle \frac{d-1}{2}}}$ for $j=1,2,\dots ,N$, we obtain

$$\parallel L_N{\left(a\right)\parallel }_{q,\kappa }\asymp m^{-{\displaystyle \frac{d-1}{q}}+{\displaystyle \frac{d-1}{2}}}{\parallel a\parallel }_{{\ell }_q^N}.$$

Therefore, for any $f\in W_{2,\kappa }^r$, we deduce

$$\begin{array}{cc}\hfill \parallel Q_N\left(f\right)-T_n{Q}_N{\left(f\right)\parallel }_{q,\kappa }& \gg \parallel P_N{Q}_N\left(f\right)-P_N{T}_n{Q}_N{\left(f\right)\parallel }_{q,\kappa }\hfill \\ & =\parallel L_N{J}_N{Q}_N\left(f\right)-L_N{J}_N{P}_N{T}_n{L}_N{J}_N{Q}_N{\left(f\right)\parallel }_{q,\kappa }\hfill \\ & \gg m^{-{\displaystyle \frac{d-1}{q}}+{\displaystyle \frac{d-1}{2}}}{\parallel J_N{Q}_N\left(f\right)-J_N{P}_N{T}_n{L}_N{J}_N{Q}_N\left(f\right)\parallel }_{{\ell }_q^N}\hfill \\ & =m^{-{\displaystyle \frac{d-1}{q}}+{\displaystyle \frac{d-1}{2}}}{\parallel y-J_N{P}_N{T}_n{L}_{N}y\parallel }_{{\ell }_q^N}.\hfill \end{array}$$

Since $g_k:={\displaystyle \frac{{\phi }_k}{\parallel {\phi }_k{\parallel }_{2,\kappa }}}$ forms an orthonormal system in $L_{2,\kappa }$ and $g_k\in H\left(v\right)=W_{2,\kappa }^{\rho }$, the random vector $(\langle f,g_1^{\left(\rho \right)}\rangle ,\dots ,\langle f,g_N^{\left(\rho \right)}\rangle )=y$ follows a standard Gaussian distribution ${\gamma }_N$ in ${\mathbb{R}}^N$. Hence, we obtain

$$\begin{array}{cc}\hfill v& \left(\left\{f\in W_{2,\kappa }^r:{∥Q_N\left(f\right)-T_n{Q}_N\left(f\right)∥}_{q,\kappa }>2c_3{m}^{\rho }{\lambda }_{n,\delta }\right\}\right)\hfill \\ \\ & \ge v\left(\left\{f\in W_{2,\kappa }^r:{\parallel y-J_N{P}_N{T}_n{L}_{N}y\parallel }_{{\ell }_q^N}>c_4{m}^{\rho +{\displaystyle \frac{d-1}{q}}-{\displaystyle \frac{d-1}{2}}}{\lambda }_{n,\delta }\right\}\right)\hfill \\ \\ & ={\gamma }_N\left(\left\{y\in {\mathbb{R}}^N:{\parallel y-J_N{P}_N{T}_n{L}_{N}y\parallel }_{{\ell }_q^N}>c_4{m}^{\rho +{\displaystyle \frac{d-1}{q}}-{\displaystyle \frac{d-1}{2}}}{\lambda }_{n,\delta }\right\}\right)\hfill \\ \\ & =:{\gamma }_N\left(G\right),\hfill \end{array}$$

where $c_4$ is a constant and $c_4$ > 0. The rank of the operator $J_N{P}_N{T}_n{L}_N$ does not exceed n, and

$${\gamma }_N\left(G\right)\le v\left(\left\{f\in W_{2,\kappa }^r:{\parallel f-T_{n}f\parallel }_{q,\kappa }>2{\lambda }_{n,\delta }\right\}\right)\le \delta .$$

Thus, we have the inequality

$${\lambda }_{n,\delta }(I_N:{\mathbb{R}}^N\to {\ell }_q^N,{\gamma }_N)\le \underset{y\in {\mathbb{R}}^N\setminus G}{sup}{\parallel y-J_N{P}_N{T}_n{L}_{N}y\parallel }_{{\ell }_q^N}\ll m^{\rho +{\displaystyle \frac{d-1}{q}}-{\displaystyle \frac{d-1}{2}}}{\lambda }_{n,\delta },$$

which leads to

$${\lambda }_{n,\delta }(W_{2,\kappa }^r,v,L_{q,\kappa })\gg n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}}}{\lambda }_{n,\delta }(I_N:{\mathbb{R}}^N\to {\ell }_q^N,{\gamma }_N).$$

This concludes the proof. □

5. Proof Main Results

The main focus of this section is to demonstrate the proof of the main theorem using some lemmas and the relevant theorems obtained in the previous section.

Proof of Theorem 1. 

It suffices to prove $S_{n,\delta }={\lambda }_{n,\delta }$, while $S_{n,\delta }=d_{\delta }^n$ follows directly from Lemma 5.

To begin with the lower bound estimation, by using Theorem 4 and Lemma 6, we can obtain the result for $1\le q<\infty$:

$$\begin{array}{cc}\hfill {\lambda }_{n,\delta }(W_{2,\kappa }^r,v,L_{q,\kappa })& \gg n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}}}{\lambda }_{n,\delta }(I_N:{\mathbb{R}}^N\to {\ell }_q^N,{\gamma }_N)\hfill \\ & \asymp n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}}}\left(N^{{\displaystyle \frac{1}{q}}}+N^{{({\displaystyle \frac{1}{q}}-{\displaystyle \frac{1}{2}})}_{+}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}\right)\hfill \\ & \asymp n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}\left(1+n^{-min\{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{q}}\}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}\right).\hfill \end{array}$$

For $q=\infty$, we have

$$\begin{array}{cc}\hfill {\lambda }_{n,\delta }(W_{2,\kappa }^r,v,L_{q,\kappa })& \gg n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}{\lambda }_{n,\delta }(I_N:{\mathbb{R}}^N\to {\ell }_q^N,{\gamma }_N)\hfill \\ & \asymp n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}\sqrt{lnN+ln{\displaystyle \frac{1}{\delta }}}\hfill \\ & \asymp n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}\sqrt{ln{\displaystyle \frac{n}{\delta }}}.\hfill \end{array}$$

The upper bound of the theorem remains to be estimated, and we now begin its proof.

For $2\le q\le \infty$ and a fixed natural number n, assume that $C_1{2}^{m(d-1)}\le n\le C_1^2{2}^{m(d-1)}$, where $C_1>0$ will be determined later. We can choose a sufficiently small $\epsilon >0$ such that $\rho >{\displaystyle \frac{d-1}{2}}+(d-1)(1+\epsilon )(\mu +\epsilon )({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})$, where $\mu$ is given in Lemma 9, and define

$$n_j=\left\{\begin{array}{cc}{u}_j,\hfill & \mathrm{if}j\le m,\hfill \\ \left[u_j{2}^{(d-1)(1+\epsilon )(m-j)-1}\right],\hfill & \mathrm{if}j>m,\hfill \end{array}\mathrm{and}{\delta }_j=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}j\le m,\hfill \\ \delta 2^{m-j},\hfill & \mathrm{if}j>m,\hfill \end{array}\right.\right.$$

where the definition of $u_j$ can be found in Theorem 3. Then, we have

$$\sum _{j\ge 0}{n}_j\ll \sum _{j\le m}{2}^{j(d-1)}+\sum _{j>m}{2}^{m(d-1)(1+\epsilon )-(d-1)\epsilon j}\ll 2^{m(d-1)},\mathrm{and}\sum _{k=1}^{\infty }{\delta }_k\le \delta .$$

It is evident that by selecting a large enough $C_1$ value, we can ensure that ${\displaystyle \sum _{j=0}^{\infty }}{n}_j\le C_1{2}^{m(d-1)}\le n$. Under Lemma 9, we know that for $\beta \in (0,{\displaystyle \frac{1}{\mu (\frac{1}{2}-\frac{1}{q})}})$, $2\le q\le \infty$, we have

$$\sum _{j=1}^{u_k}{\omega }_j^{-\beta ({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}\ll 2^{k(d-1)\left(1+\beta ({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})\right)}.$$

For $j\le m$, $n_j=u_j$, and hence ${\lambda }_{n_j,{\delta }_j}(V_j:{\mathbb{R}}^{u_j}\to {\ell }_q^{u_j},{\gamma }_{u_j})=0.$ For $j>m$, by taking ${\displaystyle \frac{1}{\beta }}=(\mu +\epsilon )({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})$ and using Lemma 7, we find that for $2\le q<\infty$, we have

$$\begin{array}{cc}\hfill {\lambda }_{n_j,{\delta }_j}(V_j:{\mathbb{R}}^{u_j}\to {\ell }_q^{u_j},{\gamma }_{u_j})& \ll 2^{j(d-1)({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})-(d-1)(1+\epsilon )(m-j)(\mu +\epsilon )({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}\hfill \\ \hfill & ·(2^{{\displaystyle \frac{j(d-1)}{q}}}+\sqrt{ln{\displaystyle \frac{2^{j-m}}{\delta }}})\hfill \\ \hfill & \ll 2^{j(d-1)({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})-(d-1)(1+\epsilon )(m-j)(\mu +\epsilon )({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}\hfill \\ \hfill & ·(2^{{\displaystyle \frac{j(d-1)}{q}}}+{(j-m)}^{{\displaystyle \frac{1}{2}}}+\sqrt{ln{\displaystyle \frac{1}{\delta }}})\hfill \\ \hfill & =I_1+I_2+I_3.\hfill \end{array}$$

(20)

and for $q=\infty$, we have

$$\begin{array}{cc}\hfill {\lambda }_{n_j,{\delta }_j}(V_j:{\mathbb{R}}^{u_j}\to {\ell }_q^{u_j},{\gamma }_{u_j})& \ll 2^{{\displaystyle \frac{j(d-1)}{2}}-{\displaystyle \frac{d-1}{2}}(1+\epsilon )(m-j)(\mu +\epsilon )}\sqrt{j+ln{\displaystyle \frac{2^{j-m}}{\delta }}}\hfill \\ \hfill & \ll 2^{{\displaystyle \frac{jd}{2}}-{\displaystyle \frac{d-1}{2}}(1+\epsilon )(m-j)(\mu +\epsilon )}(j^{{\displaystyle \frac{1}{2}}}+{(j-m)}^{{\displaystyle \frac{1}{2}}}+\sqrt{ln{\displaystyle \frac{1}{\delta }}})\hfill \\ \hfill & =I_a+I_b+I_c.\hfill \end{array}$$

(21)

We will now proceed to estimate the upper bounds for ${\lambda }_{n,\delta }(W_{2,\kappa },v,L_{q,\kappa })$. For $2\le q<\infty$, according to Equations (13) and (20), and noting that $\rho >{\displaystyle \frac{d-1}{2}}+(d-1)(1+\epsilon )(\mu +\epsilon )\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}}\right)$, we obtain

$$\begin{array}{cc}\hfill {\lambda }_{n,\delta }(W_{2,\kappa }^r,v,L_{q,\kappa })& \ll \sum _{j=m+1}^{\infty }{2}^{-j\rho }(I_1+I_2+I_3).\hfill \end{array}$$

Then, we have

$$\begin{array}{cc}\hfill \sum _{j=m+1}^{\infty }{2}^{-j\rho }{I}_1& =\sum _{j=m+1}^{\infty }{2}^{-j\rho }{2}^{j(d-1)({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})-(d-1)(1+\epsilon )(m-j)(\mu +\epsilon )({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}{2}^{{\displaystyle \frac{j(d-1)}{q}}}\hfill \\ & \ll 2^{-(d-1)(1+\epsilon )m(\mu +\epsilon )({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}\sum _{j=m+1}^{\infty }{2}^{-j(\rho -{\displaystyle \frac{d-1}{2}}-(d-1)(1+\epsilon )(\mu +\epsilon )({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}}))}\hfill \\ & \ll 2^{-(d-1)(1+\epsilon )m(\mu +\epsilon )}{2}^{-m(\rho -{\displaystyle \frac{d-1}{2}}-(d-1)(1+\epsilon )(\mu +\epsilon )({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}}))}\hfill \\ & \ll 2^{-m\rho +{\displaystyle \frac{m(d-1)}{2}}}\hfill \\ & \ll n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill \sum _{j=m+1}^{\infty }{2}^{-j\rho }{I}_2& =\sum _{j=m+1}^{\infty }{2}^{-j\rho }{2}^{j(d-1)({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})-(d-1)(1+\epsilon )(m-j)(\mu +\epsilon )({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}{(j-m)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \ll 2^{-m\rho +{\displaystyle \frac{m(d-1)}{2}}-{\displaystyle \frac{m(d-1)}{q}}}\hfill \\ & \ll n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}}}\ll n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \sum _{j=m+1}^{\infty }{2}^{-j\rho }{I}_3& =\sum _{j=m+1}^{\infty }{2}^{-j\rho }{2}^{j(d-1)({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})-(d-1)(1+\epsilon )(m-j)(\mu +\epsilon )({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}})}\sqrt{ln{\displaystyle \frac{1}{\delta }}}\hfill \\ & \ll 2^{-m\rho -{\displaystyle \frac{m(d-1)}{q}}+{\displaystyle \frac{m(d-1)}{2}}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}\hfill \\ & \ll n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q}}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}.\hfill \end{array}$$

Hence, for $2\le q<\infty$, we have

$${\lambda }_{n,\delta }(W_{2,\kappa }^r,v,L_{q,\kappa })\ll n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}(1+n^{-{\displaystyle \frac{1}{q}}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}).$$

For $p=\infty$, it follows from Equations (13) and (21), and noting that $\rho >{\displaystyle \frac{d-1}{2}}+(d-1)(1+\epsilon )(\mu +\epsilon ){\displaystyle \frac{1}{2}}$, that

$$\begin{array}{cc}\hfill {\lambda }_{n,\delta }(W_{2,\kappa }^r,v,L_{\infty ,\kappa })& \ll \sum _{i=m+1}^{\infty }{2}^{-j\rho }(I_a+I_b+I_c).\hfill \end{array}$$

Then, we have

$$\begin{array}{cc}\hfill \sum _{i=m+1}^{\infty }{2}^{-j\rho }{I}_a& =\sum _{i=m+1}^{\infty }{2}^{-j\rho }{2}^{j{\displaystyle \frac{d-1}{2}}-(1+\epsilon )(m-j)(\mu +\epsilon ){\displaystyle \frac{d-1}{2}}}{j}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \ll 2^{-m\rho +{\displaystyle \frac{m(d-1)}{2}}}{m}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \ll n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}\sqrt{lnn}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill \sum _{i=m+1}^{\infty }{2}^{-j\rho }{I}_b& =\sum _{i=m+1}^{\infty }{2}^{-j\rho }{2}^{{\displaystyle \frac{j(d-1)}{2}}-{\displaystyle \frac{d-1}{2}}(1+\epsilon )(m-j)(\mu +\epsilon )}{(j-m)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \ll 2^{-m\rho +{\displaystyle \frac{m(d-1)}{2}}}\hfill \\ & \ll n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}\sqrt{lnn}.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \sum _{i=m+1}^{\infty }{2}^{-j\rho }{I}_c& =\sum _{i=m+1}^{\infty }{2}^{-j\rho }{2}^{{\displaystyle \frac{j(d-1)}{2}}-{\displaystyle \frac{d-1}{2}}(1+\epsilon )(m-j)(\mu +\epsilon )}\sqrt{ln{\displaystyle \frac{1}{\delta }}}\hfill \\ & \ll 2^{-m\rho +{\displaystyle \frac{m(d-1)}{2}}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}\hfill \\ & \ll n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}.\hfill \end{array}$$

Hence, for $q=\infty$, we obtain

$${\lambda }_{n,\delta }(W_{2,\kappa }^r,v,L_{\infty ,\kappa })\ll n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}\sqrt{ln{\displaystyle \frac{n}{\delta }}}.$$

For $1\le q\le 2$, by the definition of ${\lambda }_{n,\delta }$, it is evident that

$${\lambda }_{n,\delta }(W_{2,\kappa }^r,v,L_{q,\kappa })\le {\lambda }_{n,\delta }(W_{2,\kappa }^r,v,L_{2,\kappa })\ll n^{-{\displaystyle \frac{\rho }{d-1}}+{\displaystyle \frac{1}{2}}}(1+n^{-{\displaystyle \frac{1}{2}}}\sqrt{ln{\displaystyle \frac{1}{\delta }}}).$$

This concludes the proof. □

6. Conclusions

Sobolev spaces play a crucial role in both theoretical research and practical applications, while weighted Sobolev classes provide theoretical guidance for addressing issues arising from non-uniform data points. This paper investigated the width problem in the weighted Sobolev space $W_{2,\kappa }^r$ on the sphere ${\mathbb{S}}^{d-1}$ under the average case and probabilistic settings, deriving the exact orders of the Gelfand and linear n-widths in both settings. These results fundamentally reveal the computational complexity and provide an effective framework for quantifying algorithm performance, offering a solid theoretical foundation for designing efficient algorithms. Meanwhile, certain aspects of the Kolmogorov width remain unresolved, and future research will focus on these open problems to further advance the study of the classical width for weighted Sobolev classes.