Probabilistic and Average Gel’fand Widths of Sobolev Space Equipped with Gaussian Measure in the S q -Norm

: In this article, we mainly studied the Gel’fand widths of Sobolev space in the probabilistic and average settings. And, we estimated the sharp bounds of the probabilistic Gel’fand ( N , δ ) -widths of multivariate Sobolev space MW r 2 ( T d ) with mixed derivative equipped with the Gaussian measure in the S q -norm by discretization methods. Later, we estimated the sharp bounds of the p -average Gel’fand N -widths of univariate Sobolev space W r 2 ( T ) and multivariate Sobolev space MW r 2 ( T d ) with mixed derivative equipped with the Gaussian measure in the S q -norm.


Introduction
When solving practical problems such as big data, artificial intelligence, engineering calculation, and signal processing, computers will produce errors due to their own defects, algorithms and calculation methods.The different definitions of cost and algorithm errors lead to different calculation models.In the classical setting, what is considered is the maximum value of the error in all possible cases.This only measures the "worst-element" case in the given class of functions.However, the optimal approximation error obtained under this algorithm is not the best approximation of most elements.In the probabilistic setting, the error is given in the "worst" case by removing the subsets of the measure at most δ.This reflects the best approximation of the "concentrated" or "most" elements in the given class of functions, and gives the distribution of the elements that reach a certain error order under a certain measure µ, depicting the intrinsic structural characteristics of the function class more accurately.In the average setting, the error is obtained by integrating the function class under a probability measure, which reflects the average level of approximation of the function class for a given measure µ.This considers the weight of each function in the function class, and reflects the essence of the function approximation problem.Therefore, it can help people more deeply understand the essential characteristics of the function approximation problem.
In recent years, width theory has attracted more and more attention.In books [1,2], we can find more detailed information about the usual widths, for example, Kolmogorov width, linear width, and Gel'fand width.Micchelli and Traub [3] expounded the relationship between computational complexity and width theory, which provided a solid theoretical foundation for solving problems such as numerical analysis and algorithm complexity.V.E.Maiorov [4,5] defined the probabilistic (n, δ)-width, and obtained the sharp-order estimate of the probabilistic Kolmogorov (n, δ)-width and linear (n, δ)-width on the space of a finite dimension equipped with the Gaussian measure by using the discretization method.Wasilkowski and Maiorov [6] studied the approximation characteristics of function classes assigned r-fold Wiener measures under the average and probability settings.Fang and Ye [7,8] studied the probabilistic linear width and the p-average linear width of the Sobolev space W r 2 equipped with Gaussian measure, and obtained the associated asymptotics in the L q -norm.Shao et al. [9] gave the concept of the probabilistic Gel'fand width.They estimated the sharp order of the probabilistic Gel'fand width of the univariate Sobolev space W r 2 equipped with Gaussian measure in the L q (T)-norm.Xu et al. [10] obtained the sharp order of the probabilistic and the p-average linear widths of the one-dimensional Sobolev space W r 2 equipped with Gaussian measure in the S q (T)-norm.With the progress and development of science and technology, research on single variables no longer satisfies the actual needs of today, so scholars have begun to step into the multivariate situation.Romanyuk [11,12] obtained the sharp-order estimates of best approximations to the classes of periodic functions of several variables by trigonometric polynomials with "numbers" of harmonics from step hyperbolic crosses.Chen and Fang [13,14] studied the multivariate Sobolev space MW r 2 with mixed derivative equipped with the Gaussian measure, obtaining the associated asymptotics of the probabilistic and the p-average Kolmogorov widths in the space L q (T d ).After that, they estimated the sharp order of the probabilistic linear (n, δ)-widths in the L q (T d )-norm.Stepanets [15] found the sharp order of the Kolmogorov widths d n (L ψ p ) p in the spaces S q (T d ) of the classes L ψ p of ψ-integrals of functions from the unit balls of S q (T d ).Dai and Wang [16] obtained the sharp order of the probabilistic and the p-average linear widths of the diagonal matrix.Liu et al. [17,18] calculated the exact order of the probabilistic and the p-average Gel'fand widths of the multivariate Sobolev space MW r 2 equipped with the Gaussian measure in the L q (T d )-norm.First, let us review some definitions.Let (X, ∥ • ∥) be the normed linear space, and denotes the deviation of the subset W from F N .Thus, e(W, F N , X) measures the extent to which the "worst" element of the subset W can be approximated from F N .

Definition 1 ([1]
).The N-widths, in the sense of Kolmogorov and linear widths, of W in X are given by d N (W, X) := inf where F N is run over all possible N-dimensional linear subspaces of X; T N is a continuous linear operator of rank not more than N from X to itself.

Definition 2 ([1]
).Let f 1 , f 2 , . . ., f N be N continuous linearly independent functionals on X such that then subspace L N of X is said to be of co-dimension N.Then, N-width, in the sense of Gel'fand, of W in X is defined by d N (W, X) := inf where the infimum is run over all linear subspaces L N of X of co-dimension N.
Definition 3 ([4-6]).Let W be the non-null subset of the space (X, ∥ • ∥), B be the Borel field on the subset W, and µ be the probabilistic measure defined on B, then µ is a σ-additive nonnegative function on B, and µ(W) = 1.For any δ ∈ [0, 1), then the corresponding probabilistic (N, δ)-widths, in the sense of Kolmogorov and linear widths, of W in X with a measure µ are given by where G δ runs over all possible subsets in B with µ(G δ ) ≤ δ.The p-average N-widths, in the sense of Kolmogorov and linear widths, of W in X are given by d Now, we introduce the concept of the probabilistic Gel'fand (N, δ)-width; let us first recall the following concept.
Let H be a Hilbert space, and be equipped with the Gaussian measure µ.Let F ⊂ H be a closed subspace, and F ⊥ be its orthogonal complement subspace.For any x ∈ H, then where the element y is called the projection of x onto F, and the decomposition form is unique.For any closed subspace G F of F such that where P F is the projection operator on F, and µ F (G F ) is the probabilistic measure on F.

Definition 4 ([9]
).Let X be the normed linear space, and be equipped with the norm ∥ • ∥.Let H be the Hilbert space, and H can be continuously embedded into X, and let µ be the probabilistic measure on the space H.Then, the corresponding probabilistic (N, δ)-width, in the sense of Gel'fand, of W in X with a measure µ is given by where L N runs over all linear subspaces of X with co-dimension not more than N, G δ runs over all possible subsets in B with µ(G δ ) ≤ δ, and which satisfies the following condition: For any closed subspace F of H, Remark 1. From ref. [9], we know that condition (2) is to ensure that there are enough elements in the set (H \ G δ ) ∩ L N .
Definition 5 ([18]).Let (X, ∥ • ∥), W, B, and µ be consistent with Definition 3.Then, the p-average N-width, in the sense of Gel'fand, of W in X is given by d N (a) (W, µ, X) p := inf where L N runs over all linear subspaces of X with co-dimension not more than N.
Next, we introduce some related symbols.Let C j > 0, j = 1, 2, . . .represent constants that are related only to the parameters r, q, ρ, and d; and, a(y) and b(y) be two arbitrary positive functions defined on the set D. Assume that there are constants C 1 , C 2 > 0, such that a(y) ≥ C 1 b(y) or a(y) ≤ C 2 b(y), y ∈ D, then let us write it as a(y) ≫ b(y) or a(y) ≪ b(y).Assume that there are constants C 3 , C 4 > 0 such that C 3 ≤ a(y) b(y) ≤ C 4 , y ∈ D, then let us write it as a(y) ≍ b(y).

Main Results
In this article, we focus on the univariate Sobolev space W r 2 (T) (d = 1), and the multivariate Sobolev space with mixed derivative MW r 2 (T d ), r = (r 1 , . . ., r d ) ∈ R d + .Now, we give their relevant definition.
Let L q , 1 ≤ q ≤ ∞, be denoted by the usual space of the q-integral of the Lebesgue space defined on and if a > s, then a j > s, j = 1, . . ., d, and Let L 2 (T d ) denote the Hilbert space, which consists of all functions that are 2π-periodic in each variable x(t) defined on T d := [0, 2π) d , and the Fourier series is given by where For any r = (r 1 , . . ., r d ) ∈ R d , we define the r-th-order derivative of x in the sense of Weyl by where We define the univariate Sobolev space W r 2 (T) by where x(0) = 1 (2π) T x(t)dt is the Fourier series of x.Then, the space W r 2 (T) is a Hilbert space, whose inner product is given by ⟨x, y⟩ r := ⟨x (r) , y (r) ⟩, x, y ∈ W r 2 (T), and the norm is given by ∥x∥ 2 1  2 − 1 q }, the normed linear space W r 2 (T) can be continuously embedded into the space L q (T), 1 ≤ q ≤ ∞.
We equip the univariate Sobolev space W r 2 (T) with a Gaussian measure µ.The mean of the Gaussian measure µ is 0, and the correlation operator C µ has eigenfunctions e k (t) and eigenvalues For any system of orthogonal functions y 1 , y 2 , . . ., y n in the space L 2 (T), and be a cylindrical subset in the space W r 2 (T).For any Borel subset B in R n , the Gaussian measure µ on G is defined by We define the multivariate Sobolev space with mixed derivative MW r where ) is a Hilbert space, whose inner product is given by ⟨x, y⟩ r := ⟨x (r) , y (r) ⟩, x, y ∈ MW r 2 (T d ), and the norm is given by ∥x∥ 2 . ., d}; we equip the multivariate Sobolev space MW r 2 (T d ) with a Gaussian measure µ.The mean of the Gaussian measure µ is 0, the correlation operator C µ has eigenfunctions e k = exp(i(k, •)) and the eigenvalues λ k = |k| −ρ , where ρ > 1.Then, For any system of orthogonal functions y 1 , y 2 , . . ., y n in L 2 (T d ), and σ j = ⟨C µ y j , y j ⟩, j = 1, . . ., n.For any Borel subset B in R n , let be a cylindrical subset in the space MW r 2 (T d ).Then, the Gaussian measure µ on G is given by We can learn more related information about the Gaussian measure µ in the papers by Kuo [19] , and Talagrand and Ledoux [20].Liu et al. [17,18] studied the Gel'fand widths, in the probabilistic and average settings, of the multivariate Sobolev space MW r 2 equipped with Gaussian measure µ in the L q -norm; the following theorem is proved by them.and δ ∈ (0, 1  2 ].Then, the Gel'fand (N, δ)-width, in the probabilistic setting, of the multivariate Sobolev space MW r 2 equipped with Gaussian measure µ in the L q -norm satisfies the asymptotics: (1) For 1 ≤ q < 2, ).
(2) For 2 ≤ q < ∞, 1  2 ], and 0 < p < ∞.Then, the Gel'fand N-width, in the average setting, of the multivariate Sobolev space MW r 2 equipped with Gaussian measure µ in the L q -norm satisfies the asymptotics: (1) For 1 ≤ q < 2, (2) For 2 ≤ q < ∞, In the normed linear space l q (1 ≤ q ≤ ∞), for x = {x j } +∞ −∞ ∈ l q , the norm is given by The following introduces the S q (T d ) space.For 1 ≤ q ≤ ∞, the definition of S q (T d ) is given by and the norm is given by ∥x∥ q,s := ∥{ x(k)}∥ l q , (11) where x(t) exp(−i(k, t))dt is the Fourier series of x.
According to Parseval equalities, we can obtain that S 2 (T d ) = L 2 (T d ).S q (T d ) ⊂ L q (T d ), if 1 ≤ q < 2; S q (T d ) ⊃ L q (T d ), if 2 < q ≤ ∞.Thus, according to Hölder inequalities, when r > max{0, 1  q − 1 2 }, the normed linear space MW r 2 (T d ) can be continuously embedded into the space S q (T d ).
Shao [21] studied the Gel'fand (N, δ)-width, in the average setting, of univariate Sobolev space W r 2 equipped with Gaussian measure µ in the S q -norm.He proved the following: Based on previous research results, this paper mainly determines the asymptotic order of the p-average Gel'fand N-width of univariate Sobolev space W r 2 (T), and the Gel'fand widths, in the probabilistic and average settings, of multivariate Sobolev space MW r 2 (T d ) equipped with the Gaussian measure µ in the S q -norm.Our main results are as follows.
Remark 2. When the dimension d = 1, the results of Theorem 5 agree with those of Theorem 3. 1  2 ], ρ > 1, and 1 < p < ∞.Then, the p-average Gel'fand N-width of MW r 2 (T d ) equipped with the Gaussian measure in the S q (T d )-norm satisfies the asymptotic q (ln (v−1)/q N).

Discretization
This paper mainly uses discretization to prove Theorem 5, the core of which is transforming the widths of the function class space to the width of the finite-dimensional space, and then, using the conclusion of the order of the probabilistic Gel'fand width in the finitedimensional space to calculate.First, we review the definition of the probabilistic Gel'fand width in the finite-dimensional space.
For x = (x 1 , x 2 , . . ., x m ) ∈ R m , where m is a positive integer.Let l m q be the normed linear space in R m with the norm We equip the finite-dimensional space R m with a standard Gaussian measure ν = ν m , which is defined by for any Borel subset and ν(R m ) = 1.
Let N be a non-negative integer, then for any δ ∈ [0, 1), the Gel'fand (N, δ)-width, in the probabilistic setting, of the finite-dimensional space R m equipped with the standard Gaussian measure ν in l m q -norm is defined by where L N runs over all linear subspaces of R m with co-dimension not more than N, and G δ runs over all possible subsets in B with ν(G δ ) ≤ δ.Tan et al. [9] obtained the Gel'fand (N, δ)-width of the finite-dimensional space in the probabilistic setting, the result is as follows: Lemma 1 ([9]).Let 2N ≤ m, for any δ ∈ (0, 1  2 ], then: (1) For 1 ≤ q < 2, (2) For 2 ≤ q < ∞, In order to establish the discretization theorem better, the following two special Borel sets in the finite-dimensional Spacesit are of great importance.

Lemma 4 ([9]
).Let (X, ∥ • ∥) be the normed linear space; H be the Hilbert space, and H can be embedded continuously into X; and µ be the probabilistic measure on H, δ ∈ (0, 1  2 ].Then, Lemma 5 ([22]).Let r = (r 1 , . . ., r d ) ∈ R d , which satisfies the condition , Then, the probabilistic linear width of MW r 2 equipped with the Gaussian measure in the space S q (T d ) satisfies the asymptotics From Lemmas 4 and 5, the lower bound of Theorem 5 is easily proved.Thus, we only need to use the discretization method to establish the upper bound of Theorem 5. Next, to facilitate the computation, it is necessary to introduce some notation and split the Fourier series of the function into the sum of diadic blocks.
For any s = (s 1 , s 2 , . . ., s d ) ∈ N d , let . ., d .Let δ s x(t) denote the "block" of the Fourier series for x(t), namely, where k ≥ d, and c n e i(n,•) , is an isomorphic mapping from the space of trigonometric polynomials span e i(n,•) : n ∈ M s to the space R 2 (s,1) .
According to Lemma 7, we can obtain Now, for any l, k, we consider a mapping Therefore, it follows from Lemma 6 and Equation ( 18) that I l,k is a linear isomorphic mapping from F l,k to l ∥S l,k ∥ q . From Equation ( 9), we can obtain Based on the above analysis, the discretization theorem for the upper bound of Theorems 5 can be obtained as follows.
Theorem 7.For 1 ≤ q < ∞, let r = (r 1 , . . ., r d ) ∈ R d , which satisfies the condition 1  2 ].Assume that there exist the sequences of numbers {N l,k } and {δ l,k }, which satisfy the conditions 0 ≤ N l,k ≤ ∥S l,k ∥, ∑ l,k N l,k ≤ N, and Proof.By Lemma 1, assume that there exists a constant c ′ > 0 such that , where c 0 and c q are the same as Lemmas 2 and 3.
From ref. [9], we have ∥x∥ From Lemmas 2 and 3, we can obtain ν(Q l.k ) ≤ δ l,k .Obviously, for any linear subspace Let L l,k be subspaces of R ∥s l,k ∥ with co-dimension at most N l,k , Therefore, From Equation ( 20), there exists a constant c 1 > 0, such that Consider the subset of MW r 2 (T d ) From the definition of the Gaussian measure µ and the standard Gaussian measure ν in the finite-dimensional space R ∥S l,k ∥ , by virtue of Equations ( 19) and ( 21), then G l,k ; according to the assumptions of the theorem, we can obtain is the direct sum.Then, for any linear subspace is the direct sum.
According to the definition of the Gel'fand (N, δ)-width in the probabilistic setting, we can obtain

Proofs of Theorem 4
In this part, we mainly prove the Gel'fand N-width, in the average setting, of the univariate Sobolev space W r 2 (T).This mainly uses the results of the probabilistic Gel'fand (N, δ)-width and the method of real analysis to estimate the p-average Gel'fand N-width.It is known that any subspace of L 1 (T) is absolutely convergent in the sense of l q with respect to the Fourier coefficients.
Proofs of Theorem 4. We calculate the asymptotic order in two cases.Let F N be a linear subspace of S q (T) with co-dimension at most N. Consider the set sequence {G 2 −k } ∞ k=0 , which satisfies µ(G 2 −k ) ≤ 2 −k , for any k, and G 1 = W r 2 (T), for k = 0. (1) For 1 ≤ q < 2, according to the result of Theorem 3, we have According to the definition of Gel'fand N-width in the average setting and estimate (22), it can be known that Now, we start to prove the lower bound of Theorem 4. From Theorem 3, let C > 0 be a constant, then Consider the set Then, µ(G) > 1 2 ; otherwise, µ(G) ≤ 1 2 .According to the definition of the Gel'fand (N, δ)width in the probabilistic setting, we can obtain This contradicts the inequality (23), consequently, µ(G) > 1 2 .Therefore, (2) For 2 ≤ q < ∞, according to the result of Theorem 3, we have According to the definition of Gel'fand N-width in the average setting and estimate (24), it can be known that Now, we start to prove the lower bound of Theorem 4. From Theorem 3, let C > 0 be a constant, then Consider the set Then, µ(G) > 1 2 .This contradicts inequality (25), consequently, µ(G) > 1 2 .Therefore, By synthesizing the proofs and results of (1) and (2), we can obtain Theorem 4 is proved.

Proofs of Theorem 5
Regarding the proof of the upper bound of Theorem 5, we need to recall two important lemmas: Lemma 8. Let N be a non-negative integer, and N ≍ 2 u u v−1 , β > 0, S l,k be defined as Formula (15) where ⌊a⌋ is the largest integer no greater than a; and β is a constant, which satisfies also the condition 0 Lemma 9.For N ∈ N, suppose that there exists u satisfying condition N ≍ 2 u u v−1 .Let According to Lemmas 8 and 9, we can obtain ∑ l,k δ l,k ≤ δ.Then, the sequences of numbers {N l,k } and {δ l,k } satisfy the conditions of Theorem 6.
Theorem 6 is proved.

Summary
In this article, we mainly studied the Gel'fand widths, in the probabilistic and average settings, of Sobolev space in the S q -norm by discretization methods.As we know, with the development of data science, mathematical theory plays an increasingly important role in big data research.The complexity of algorithms, the selection of optimal algorithms, and the convergence speed of approximation algorithms have always been important research directions in big data science.This research, by the width of the theory, is the effective method to solve such problems.
However, this study has some limitations, for example, we did not discuss the case q = ∞, which is also worth investigating later.In addition, expanding the research scope of function classes and studying the characteristics of nonlinear function approximation in different frameworks will also be a direction we need to study in the future.