Gel’fand-N-width in probabilistic setting

In this article, we first put forward a new definition of probabilistic Gel’fand-(N,δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(N,\delta )$\end{document}-width which is the classical Gel’fand-N-width in a probabilistic setting. Then we estimate the sharp order of the probabilistic Gel’fand-(N,δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(N,\delta )$\end{document}-width of finite-dimensional space. Furthermore, we obtain the exact order of probabilistic Gel’fand-(N,δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(N, \delta )$\end{document}-width of univariate Sobolev space by the discretization method according to the result of finite-dimensional space.

In order to present the relevant results, we first introduce some notations. Let R, Z and N denote the real number set, the integer set and the nonnegative integer set, respectively. Assume that c, c i , c i , i = 0, 1, . . . , are positive constants depending only on the parameters p, q, r, ρ. The notation a(y) b(y) or a(y) b(y) for two positive functions a(y) and b(y) means that there exist constants c, c 1 and c 2 such that c 1 ≤ a(y)/b(y) ≤ c 2 or a(y) ≤ cb(y) for any y ∈ D.

Gel'fand-N-width in a probabilistic setting
In this section, we first review notions of the width in worst case setting, then introduce the definitions of probabilistic Kolomogorov-(N, δ)-width and probabilistic linear-(N, δ)width. Finally, we propose the new definition of the probabilistic Gel'fand-(N, δ)-width.

Definition 2.1
Suppose that X is a normed linear space equipped with a norm · , W is a non-null subset of X, N is nonnegative integer. Then are called Kolomogorov-N -width, linear N -width, Gel'fand-N -width of W in X, respectively. Here, F N runs through all linear subspaces of X with dimension at most N , T N runs through all bounded linear operators on X with rank at most N , L N runs all linear subspaces of X with codimension at most N .
The codimension is defined as follows. A linear subspace L N of the normed linear space X is called codimension N , if there are N linear independent continuous linear functionals f 1 , f 2 , . . . , f N on X, such that We simply write inf ∅ = +∞. Definition 2.2 Let (X, · X ) and (Y , · Y ) be normed linear spaces, and X can be imbedded continuously into Y . Then are regarded as Kolomogorov-N -width, linear N -width and Gel'fand-N -width of X in Y , respectively. Here, T N is taken over all continuous linear operators from X into Y of rank at most N , L N is taken over all subspaces of X of the codimension at most N , B(X) is the unit ball of X.
There are a series of useful properties of the width in worst case setting which are as follows.
Proposition 2.1 ([2]) Let X and Y be normed linear spaces, and X can be imbedded continuously into Y . Then More details about the width in worst case setting can be consulted in [2]. Now, we tend to introduce the width in a probabilistic setting. Maiorov [3,4] has proposed the definitions of Kolomogorov-N -width and linear N -width in a probabilistic setting.

Definition 2.3
Suppose that X is a normed linear space, W ⊂ X, and B is a Borel field formed by all opened sets in W , μ is a probabilistic measure on B, δ ∈ (0, 1], then are called probabilistic Kolomogorov-(N, δ)-width and probabilistic linear-(N, δ)-width of W in X with measure μ, respectively. Here G δ is taken over all subsets in B with the measure at most δ.
Remark 2.1 Comparing Definition 2.1 with Definition 2.3, we can obviously find that the N -width in a probabilistic setting is just the N -width in a worst case setting by eliminating the subset whose measure is at most δ.
In order to define the Gel'fand-N -width in a probabilistic setting, we first introduce the following obvious result.
Let H be a Hilbert space with the probabilistic measure μ, F be a closed subspace in H. Let F ⊥ denote the orthogonal complement of F. Any x ∈ H can be decomposed uniquely in the form The element y will be denoted by Px and P is called projection operator upon F. For any Borel set G F in F, let Then μ F is a probabilistic measure on F.
With Definitions 2.1, 2.2, 2.3 and Remark 2.1, we now can put foreword the new definition of Gel'fand-N -width in a probabilistic setting.

Definition 2.4
Let H be a Hilbert space, (X, · ) be a normed linear space, and H can be imbedded continuously into X, μ be a probabilistic measure on H, δ ∈ (0, 1]. Then Here L N runs over all linear subspaces of H with codimension at most N . G δ is taken over all subsets of H with the measure at most δ, and satisfies the following condition: for any closed subspace F in H, Here μ F refers to Eq. (1).
Remark 2.2 Here we add the condition (2) in order to make sure that (H\G δ ) ∩ L N has enough elements.
There is a very useful relationship between the probabilistic linear-(N, δ)-width and the probabilistic Gel'fand-(N, δ)-width.

Theorem 2.1
Suppose H is a Hilbert space, (X, · ) is a normed linear space, H can be imbedded continuously into X, δ ∈ (0, 1], μ is a probabilistic measure on H. Then Proof ∀ε > 0, by the definition of d N δ (H, μ, X), there is subspace L N of H with codimension at most N , and subset Q ⊂ H, for which Then T N is a bounded linear operator from H into X with the rank at most N .
It is clear that Since ε is arbitrary, one has

Gel'fand-N-width of finite-dimensional space in a probabilistic setting
In this section, we will discuss the Gel'fand-N -width of finite-dimensional space in a probabilistic setting. We first review the finite-dimensional space.
We denote by B m p (ρ) := {x ∈ l m p : x l m q ≤ ρ} a ball with the radius ρ in l m q , and B m p := B m p (1). It is clear that l m 2 is a Hilbert space with inner product There are several useful results about linear width and Gel'fand width of finitedimensional space in worst case setting as follows.
Considering the space of l m 2 with the standard Gaussian measure γ = γ m , which is defined as where G is any Borel subset in l m 2 . Obviously, γ (l m 2 ) = 1. Maiorov, Fang Gensun and Ye Peixin have obtained the sharp order of linear width of finite-dimensional space in probabilistic setting which can be summarized as follows. 1 2 ]. Then: Here the upper bounds only need the condition of N ≤ m.
For discussing the sharp order of Gel'fand-(N, δ)-width of finite-dimensional space, we introduce two special Borel sets in finite-dimensional space.
We now start to estimate the exact order of the Gel'fand-(N, δ)-width of finite-dimensional space. 1 2 ]. Then: Here the upper bounds hold if N ≤ m.
Proof It is obvious that the lower bound holds by the Theorem 2.1 and Theorem 3.2. Now,we estimate the upper bound.
where c 0 is the constant of Lemma 3.1. By Lemma 3.1, γ (G δ ) ≤ δ. It is clear that G δ which is the subset in l m 2 satisfies condition in Definition 2.4. From the definition of the Gel'fand-(N, δ)-width, we have (2) For 2 ≤ q < ∞.
where c q is the constant of Lemma 3.2. By Lemma 3.2, γ (G δ ) ≤ δ. Obviously, G δ also satisfies the condition of Definition 2.4. Using the definition of Gel'fand-(N, δ)-width, we have

Gel'fand-(N, δ)-width of univariate Sobolev space
In this section, we estimate the exact order of Gel'fand-(N, δ)-width of univariate Sobolev space. Denote by L q (T), 1 ≤ q ≤ ∞, the classical q-integral Lebesgue space of 2π -periodic functions with the usual norm, · L q := · L q (T) . It is clear that L 2 (T) is a Hilbert space with inner product For any x ∈ L 2 (T), the Fourier series of x can be regarded as wherex(k) = 1 2π T x(t)e ikt dt, e k (t) = e ikt .
For arbitrary r ∈ R, we define the rth order derivative of x in the sense of Weyl by where (ik) r = |k| r e iπ r 2 sgn k . Let It is well-known that W r 2 (T) is a Hilbert space with the inner product x, y r := x (r) , y (r) , x, y ∈ W r 2 (T), and the norm can be obtained By the Parseval equality, we can obtain W r 2 (T) is named as univariate Sobolev space. It is well-known that W r 2 (T) (r > 1 2 ) can be embedded continuously into the L q (T), 1 ≤ q ≤ ∞. Numerous approximation characteristic of the univariate Sobolev space, such as Kolomogorov-N -width, linear N -width in worst case setting, probabilistic setting and average setting, have been referred to in the literature. Now we equipped W r 2 (T) with a Gaussian measure μ whose mean is zero and correlation operator C μ has eigenfunctions e k (t) and eigenvalues That is, Let y 1 , . . . , y n be any orthogonal system of functions in L 2 (T), σ j = C μ y j , y j , j = 1, . . . , n, and B be an arbitrary Borel subset of l n 2 . Then the Gaussian measure μ on the cylindrical subset G in the space W r 2 (T) is given by More detailed information about the Gaussian measure in Banach space can be found in the books by Kuo [11], Ledoux and Talagrand [12]. Maiorov, Fang Gensun and Ye Peixin have researched the exact order of the Kolomogorov- (N, δ)-width and the linear-(N, δ)-width of univariate Sobolev space W r 2 (T).
We now discuss the order of Gel'fand-(N, δ)-width of W r 2 (T) in L q (T). It is the main result of this section.
According to Theorem 2.1 and Theorem 4.1, we can easily obtain the lower bound of Theorem 4.2. Now, we just need to estimate the upper bound of Theorem 4.2 by the discretization method. At first, we introduce some notations and results.
For natural number k, set It is obvious that where |A| denotes the cardinality of A. For arbitrary natural number k and x = n∈Z 0 c n e int , denote and k x(t) = n∈S k c n e int .
For arbitrary x ∈ F k , by Lemma 4.1, we have In order to establish the discrete theorem to estimate the upper bound, we consider the polynomial in the space F k We have Plugging this into Eq. (5), we obtain For any k ∈ N, we consider the following mapping: By (6), I k is linear isomorph from the space F k into the space l Then there is c > 0 such that σ = c 2 -k(ρ-1) . Theorem 4.3 Suppose 1 < q < ∞, r > 1 2 , δ ∈ (0, 1 2 ], N ∈ N. Let {N k } and {δ k } be nonnegative integer sequence and nonnegative real sequence, respectively, in which k∈Z N k ≤ N and Proof For any k ∈ N. From Theorem 3.3, there is a positive constant c q such that For c 0 and c q see Lemma 3.1 and Lemma 3.2, respectively. By and Theorem 3.3, we can conclude that if y ∈ Q k , then From Lemma 3.1 and Lemma 3.2, we have It is clear that Q k satisfies the condition (2) in Definition 2.4. Let L k be a subspace in l |S k | 2 with codimension at most N k . Then For any x ∈ W r 2 (T), by (5), there is a constant c > 0 such that Consider the set of W r 2 (T) Then It is clear that the subspace F k := D -r I -1 k L k has codimension at most N k in W r 2 (T) and Consider the set G = ∞ k=1 G k and the subspace F N = k F k ⊂ W r 2 (T), where the sum is a direct sum. We obtain By the definition of the Gel'fand-(N, δ)-width and (8), we have In order to estimate the upper bound of Theorem 4.3, we also need the following lemma.  Proof According to Theorem 2.1 and Theorem 4.1, we can easily obtain the lower bound of Theorem 4.2. We now just need to prove the upper bound.