N -Widths of Multivariate Sobolev Spaces with Common Smoothness in Probabilistic and Average Settings in the S q Norm

: In this article, we give the sharp bounds of probabilistic Kolmogorov ( N , δ ) -widths and probabilistic linear ( N , δ ) -widths of the multivariate Sobolev space W A 2 with common smoothness on a S q norm equipped with the Gaussian measure µ , where A ⊂ R d is a ﬁnite set. And we obtain the sharp bounds of average width from the results of the probabilistic widths. These results develop the theory of approximation of functions and play important roles in the research of related approximation algorithms for Sobolev spaces


Introduction
The approximation theory of functions is a classical theory of basic mathematics and computational mathematics, and width theory plays a very important role in approximation theory.With the gradual development of modern mathematics and science, the system of width theory has also been improved, which has greatly promoted the research of algorithms and computational complexity.Different types of widths correspond to different calculation methods, and then result in different errors.The different definitions of algorithm errors and costs lead to different computational models.The most common models are the worst case setting, probabilistic setting, and average-case setting.Temlyakov [1] calculated the bounds of approximation of functions with a bounded mixed derivative.Maiorov [2][3][4] gave the definition of probabilistic Kolmogorov and linear (N, δ)-widths and obtained the sharp bounds of probabilistic Kolmogorov (N, δ)-widths of Sobolev space W r 2 in L q by using discretization.Fang and Ye [5,6] estimated the exact order of linear N-widths in the probabilistic setting and average-case setting of finite dimensional space.Chen and Fang [7,8] discussed probabilistic Kolmogorov (N, δ)-widths and probabilistic linear (N, δ)-widths of the multivariate Sobolev space MW r 2 T d with a mixed derivative, and they obtained the sharp bounds of p-average Kolmogorov and linear N-widths of MW r 2 T d .Tan et al. [9] gave the definition of probabilistic Gel'fand (N, δ)-width and obtained the sharp bounds of probabilistic Gel'fand (N, δ)-width of Sobolev space W r 2 (T).Liu et al. [10] gave the definition of p-average Gel'fand N-width and obtained the sharp bounds of p-average Gel'fand N-widths of Sobolev space W r 2 (T) and MW r 2 T d .Dai and Wang [11] obtained the sharp bounds of probabilistic linear (N, δ)-widths and p-average linear N-widths of finite dimensional space with a diagonal matrix.Wang [12,13] estimated the sharp bounds of probabilistic linear (N, δ)-widths and p-average linear N-widths of weighted Sobolev spaces on the ball and Sobolev spaces on compact two-point homogeneous spaces.
Let us recall some definitions of N-widths, which can be found from the book of Pinkus [14].
Let W be a bounded subset of a normed linear space X with norm • , and F N be a N-dimensional subspace of X.The following quantity is called the deviation of W to F N : E(W, F N , X) := sup x∈W e(x, F N ).
where e(x, F N ) := inf y∈F N x − y .It shows how well the "worst" elements of W can be approximated by F N .And the Kolmogorov N-width of W in X is defined as follows: where the leftmost infimum is taken over all N-dimensional linear subspaces of X.
Next, let T be a linear operator from X to X.The linear distance of the image TW from the set W is defined as follows: and the linear N-width of W in X is defined as follows: where the infimum is taken over all linear operators T N whose rank is at most N. Now we give the definition of probabilistic (N, δ)-widths and p-average N-widths from the article of Maiorov [2][3][4].

Definition 1.
Let W be a bounded subset of normed linear space (X, • ).Assume that W contains a Borel field B consisting of open subsets of W and is equipped with a probability measure µ, i.e., µ is a σ-additive nonnegative function on B , and satisfies the condition that µ(W) = 1.For any δ ∈ (0, 1], the probabilistic Kolmogorov (N, δ)-width and probabilistic linear (N, δ)-width of W in X are defined as follows: (3) where G δ runs through all possible subsets in B, which satisfies the condition that µ(G δ ) ≤ δ.
Definition 2. Let W, X and µ be the same to Definition 1.Given 0 < p < ∞, the p-average Kolmogorov N-width and p-average linear N-width are defined, respectively, by N (W, µ, X) p := inf It can be seen from the definition that N-widths are defined by the errors generated by the "worst" elements of the functions class during the approximation process in the worst case setting.For example, the classical Kolmogorov N-widths of functional classes are defined by the optimal errors generated by the approximation of the "worst" element in the set by a finite dimensional subspace.To satisfy the demands of practical applications and theoretical analysis, the concepts of N-widths in the probabilistic and average-case setting are introduced.The sharp bounds of those widths are often used to solve the optimal solution of numerical problems.Like classical N-widths, probabilistic (N, δ)-widths reflect the best approximation of functional classes.From the definitions, we know that it needs to delete some functions with the "worst" properties before defining N-widths of functional classes in the probabilistic setting, and these widths are still defined by the "worst" elements of the remaining functions.Therefore, although the probabilistic (N, δ)-widths can allow the algorithm to generate "errors" within a given range, it does not reflect the overall optimal approximation situation.The N-widths in the average-case setting are defined by the integral of the errors under a given measure, which give the average approximation degree of a function class under a given probability measure.They reflect the optimal approximation degree of most elements in spaces, and more profoundly reflect the essential characteristics of the structure of the functional classes.
Next, we will provide two asymptotic relationships.Let a(x) and b(x) be two positive functions of x.If there is a positive constant c > 0, such that a(x) ≤ cb(x) for all x from the domain of the functions a and b, then we write a(x) b(x) or b(x) a(x).If a(x) b(x) and a(x) b(x), then we write a(x) b(x).

Main Results
In this article, we will discuss probabilistic Kolmogorov and linear (N, δ)-widths.Then, we will estimate the sharp bounds of p-average Kolmogorov and linear N-widths by using the results of probabilistic Kolmogorov and linear (N, δ)-widths.First, we introduce the concept of multivariate Sobolev space Assume L 2 T d is a classical Lebesgue square integrable space.For any x, y ∈ L 2 T d , this space is a Hilbert space with the inner product For x ∈ L 2 T d , the Fourier series of x is defined as follows: where e k (t) := exp i(k, t).
For any α = (α 1 , . . . ,α d ) ∈ R d , we define the Wyel α-derivative for x ∈ L 2 T d as follows: Given the finite subset A of R d , the multivariate Sobolev W A 2 T d with common smoothness is defined by x(t)dt j = 0, j = 1, . . ., d .(7) From Equation (7), we need to give the definition of the common Weyl-derivative as follows: where (ik) A = ∑ α∈A (ik) α .We can know that the Sobolev space W A 2 T d is a Hilbert space with the inner x, y A := x (A) , y (A) and with the norm Our results of the Sobolev space W A 2 T d with common smoothness can be a generalization of the sharp bounds of N-widths in the probabilistic and average setting of Sobolev spaces with smoothness.For example, if 2 T d is a Sobolev space with a mixed derivative, and the related conclusions can be found in papers [7,8].
We denote by A and B any two subsets of R d , and we denote that Let co(A) be the convex hull of a set A, N(A) := co(A) − R d + , and I N(A) be the set of interior points of N(A).We write A 0 In the research process of this article, we always assume that 0 ∈ I N(A) and r > 1/2.Now, we give the definition of the space S q T d : where l q is the infinite vector space with the norm for any a = {a n } ∞ n=−∞ : Next, we equip a Gaussian measure for W A 2 T d .Let µ be a Gaussian measure whose mean value is 0 and whose correlation operator is C µ which has eigenfunctions e k (t) and eigenvalues Let y 1 , . . ., y n be any orthogonal system of functions in L 2 T d , σ j = C µ y j , y j , j = 1, . . ., n, and D be an arbitrary Borel subset of R n .Then, the Gaussian measure µ on the cylindrical subsets in the space W A 2 T d : More results and research of Gaussian measures can be found in paper [15][16][17].
The aim of this paper is to determine the asymptotic order of probabilistic Kolmogorov and linear (N, δ)-widths as well as p-average Kolmogorov and linear N-widths of the multivariate Sobolev space W A 2 T d with common smoothness.The main results are as follows:

Discretization
In order to prove Theorems 1 and 2, we use the discretization method, which is based on the reduction of the calculation of the probabilistic widths of a given class to the computation of the widths of a finite-dimensional set equipped with the standard Gaussian measure.Before we use the discretization, we need the definitions, and cite some results on the probabilistic widths of finite-dimensional spaces.Let l m p be the m-dimensional normed space of vectors Consider in R m the standard Gaussian measure, which is defined as where First, we introduce some results of probabilistic (N, δ)-widths of finite space.These results can be found from papers of Maiorov, Chen, Fang, and Ye [2][3][4][5][6][7][8].
Lemma 1 (Maiorov, Chen, and Fang [3,4,7]).Let m > N, 1 ≤ q ≤ ∞ and δ ∈ 0, 1  2 .Then, Lemma 2 (Maiorov, Fang, and Ye [2,5,6]).Let m > N, 1 ≤ q ≤ ∞ and δ ∈ 0, 1 2 .Then, Lemma 3 (Maiorov [3]).For ∀δ ∈ 0, 1  2 , there is a positive c 0 , such that For 2 ≤ q < ∞ and any δ ∈ 0, 1  2 , there exists a positive constant c q , which depends only on the q, such that To establish the discretization theorem, we introduce some notations and lemmas.It is convenient in many cases to split the Fourier series of a function into the sum of diadic blocks.We associate every vector s = (s 1 , • • • , s d ) ∈ N d whose coordinates are natural numbers with the set And we let x s (t) be the "block" of the Fourier series for x(t), denoted by After introducing these necessary concepts, we have Lemma 4 (Galeev [18]).Let s ∈ N d .Then, the trigonometric polynomial space span{e n (t) : n ∈ s } and R 2 (s,1) are isomorphic under the following mapping: c n e n (t), For natural numbers l and k, we define where S A (s) = sup{(s, α) : α ∈ A}.We can know k ≥ d, and We can obtain that S l,k = 2 k S l,k .And we define ∆ l,k x := ∑ s∈S l,k δ s x.
From ( [7]), for any α ∈ A, n α 2 (s,α) .So, From the definition of A , we know Therefore, for any x ∈ F l,k , we have That is, We consider a mapping: .
It is not difficult to see that I l,k is a isomorphic mapping.From Equation (9), we know that By |n| 2 (s,1) = 2 k ([7]), we obtain Therefore, from Equation (25), we have Based on the above description, we establish the discretization theorem.The following theorems reflect the upper bounds of Theorems 1 and 2. 1  2 , N ∈ N, A satisfy the condition of Theorem 1. Assume that the sequences of numbers N l,k and δ l,k satisfy the condition Proof.From Definition 1, there would be a subspace L l,k of l where From Equation (27), there is a constant c 1 > 0 independent of l and k, such that From Equations ( 29) and (30), the definition of µ and v, Consequently, by Definition 1, we have which completes the proof of Theorem 5.
To estimate the upper bound of Theorem 1, we need the following lemmas.
Lemma 5 (Romanyuk [19]).Assume that the set A satisfies the condition of Theorem 1, then where u ∈ R + .
From Lemma 5, we have where a is the integer part of a.Then, we can choose c, such that N l,k ≤ S l,k , ∑ l,k N l,k ≤ N.
To establish the discretization of the lower bound of Theorem 1, we also need the following concepts.Let where the constants c 0 and k are pending.Then, Let F S = span{e n (t) : n ∈ s , s ∈ S}.Consider the mapping: .
Then for any x ∈ F S , by using the method of the proof of Equation ( 26), we can obtain 2 (r+ρ/2)k x q,s . (32) Proof.From Definition 1, there is a subspace F 1 , such that dim F 1 ≤ N and where Equation ( 35) can be obtained by Equation (32); therefore, Due to dim I S D A F 1 = dim F 1 = N and Definition 1, we have 1  2 , N ∈ N, A satisfy the condition of Theorem 1. Assume that the sequences of numbers N l,k and δ l,k satisfy the condition Proof.From Definition 1, there would be a linear operator T l,k of l where From Equation ( 27), there is a constant c 3 > 0 independent of l and k, such that Consider the set From Equations (37) and (38), the definition of µ and v, Consequently, by Definition 1, we have which completes the proof of Theorem 7.
Proof.From Definition 1, there is a linear operator T 1 , such that rankT 1 ≤ N and where Equation ( 41) can be obtained by Equation (32).Let Due to rankI S D A T 1 = rankT 1 = N and Definition 1, we have

Proof of Main Results
Now we prove Theorem 1 by using Theorems 5 and 6 and Lemma 1, and prove Theorem 2 by using Theorems 7 and 8 and Lemma 2. And then, we prove Theorems 3 and 4 by using results of Theorems 1 and 2. Assume that N l,k satisfies the condition of Lemma 5 and assume that N ∈ N satisfies the condition N 2 u u v .Let Proof of Theorem 1. From Theorem 5, Lemma 1, for 1 ≤ q < 2, we have First, we calculate I 1 : where ∑ is carried out over k for S l,k ≤ l v , and ∑ is carried out over k for S l,k > l v .Therefore, ∑ d≤k≤l S l,k 1/q 2 k/q ≤ l v/q ∑ d≤k≤l 2 k/q l v/q 2 l/q ; and ∑ d≤k≤l S l,k 2 (s,1)/q l v/q 2 l/q .Therefore, ∑ d≤k≤l S l,k 1/q 2 k/q l v/q 2 l/q .(42) So, we obtain Due to 0 < β < min{2r + ρ − 2, 1/2}, we have Secondly, we calculate I 2 : By using the method of the proof of Equation ( 42), we can obtain Therefore, Due to 0 < β < min{2r + ρ − 2, 1/2}, we have Finally, we calculate I 3 : By using the method of the proof of Equation (42), we can obtain ∑ d≤k≤l S l,k 1/q−1/2 2 k/q−k/2 l v/q−v/2 2 l(1/q−1/2) . Therefore, 2 −(r+ρ/2−1/q)u−u/2 u v/q−v/2 ln(1/δ) If 2 ≤ q < ∞, from Theorem 5, Lemma 1, and the definition of N l,k , we have First, we calculate I 1 : By using the method of the proof of Equation (42), we can obtain ∑ d≤k≤l S l,k 1/q 2 k(1/q+1/2−β/2) l v/q 2 l(1/q+1/2−β/2) .Therefore, Due to 0 < β < min{2r + ρ − 2, 1/2}, we obtain Secondly, we calculate I 2 : By using the method of the proof of Equation ( 42), we can obtain Therefore, Finally, we calculate I 3 : By using the method of the proof of Equation (42), we can obtain Therefore, Proof of Theorem 2. First, we prove the upper bound of Theorem 2. From Lemma 2, if have the same sharp bounds.So, we only need to prove the upper bound if 2 ≤ q < ∞.From Theorem 7 and Lemma 2, we obtain It is obvious to see that I 1 = I 1 .Therefore, Now, we calculate I 2 : By using the method of the proof of Equation (42), we can obtain Therefore, Due to the astringency of r+ρ/2−1/q ln v/q N , 1 ≤ q < ∞.
Next, we prove the lower bound of Theorem 4. We consider the set G = x ∈ W A 2 T d : x − T N x q,S > 1 2 λ N,1/2 W A 2 T d , µ, S q T d .

2 T
d , µ, S q T d .So, we obtain contradictions.Therefore,W A 2 (T d ) e x, F N , S q T d p dµ G e x, F N , S q T d p dµ G 1 2 d N,1/2 W A 2 T d , µ, S q T d p dµ 2 −p N −1 ln v N