Probabilistic linear widths of Sobolev space with Jacobi weights on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[-1

Optimal asymptotic orders of the probabilistic linear \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}(n,δ)-widths of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda_{n,\delta }(W^{r}_{2,\alpha,\beta }, \nu,L_{q,\alpha,\beta })$\end{document}λn,δ(W2,α,βr,ν,Lq,α,β) of the weighted Sobolev space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W_{2,{\alpha, \beta }}^{r}$\end{document}W2,α,βr equipped with a Gaussian measure ν are established, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{q,\alpha,\beta }$\end{document}Lq,α,β, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq q\leq \infty $\end{document}1≤q≤∞, denotes the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{q}$\end{document}Lq space on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[-1,1]$\end{document}[−1,1] with respect to the measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(1-x)^{\alpha }(1+x)^{\beta }$\end{document}(1−x)α(1+x)β, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha,\beta > -1/2$\end{document}α,β>−1/2.


Introduction
This paper mainly focuses on the study of probabilistic linear (n, δ)-widths of a Sobolev space with Jacobi weights on the interval [-, ]. This problem has been investigated only recently. For calculation of probabilistic linear (n, δ)-widths of the Sobolev spaces equipped with Gaussian measure, we refer to [-]. Let us recall some definitions.
Let K be a bounded subset of a normed linear space X with the norm · X . The linear n-width of the set K in X is defined by where L n runs over all linear operators from X to X with rank at most n.
Let W be equipped with a Borel field B which is the smallest σ -algebra containing all open subsets. Assume that ν is a probability measure defined on B. Let δ ∈ [, ). The probabilistic linear (n, δ)-width is defined by where G δ runs through all possible ν-measurable subsets of W with measure ν(G δ ) ≤ δ. Compared with the classical case analysis (see [] or []), the probabilistic case analysis, which reflects the intrinsic structure of the class, can be understood as the ν-distribution of the approximation on all subsets of W by n-dimensional subspaces and linear operators with rank n.
In his recent paper [], Wang has obtained the asymptotic orders of probabilistic linear (n, δ)-widths of the weighted Sobolev space on the ball with a Gaussian measure in a weighted L q space. Motivated by Wang's work, this paper considers the probabilistic linear (n, δ)-widths on the interval [-, ] with Jacobi weights and determines the asymptotic orders of the probabilistic linear (n, δ)-widths. The difference between the work of Wang and ours lies in the different choices of the weighted points for the proofs of discretization theorems.

Main results
Consider the Jacobi weights where h n (α, β) = (α + β + ) (α + ) (β + ) (n + α + ) (n + β + ) (n + α + β + ) (n + ) (n + α + β + ) ∼ n - with constants of equivalence depending only on α and β. Then the normalized Jacobi polynomials P n (x), defined by form an orthonormal basis for L ,α,β , where the inner product is defined by Denote by S n the orthogonal projector of L  (w α,β ) onto n in L  (w α,β ), which is called the Fourier partial summation operator. Consequently, for any f ∈ L  (W α,β ), It is well known that (see Proposition .. in []) P (α,β) n is just the eigenfunction corresponding to the eigenvalues -n(n + α + β + ) of the second-order differential operator Given r > , we define the fractional power (-D α,β ) r/ of the operator -D α,β on f by in the sense of distribution. We call f (r) := (-D α,β ) r/ the rth order derivative of the distribution f . It then follows that for f ∈ L ,α,β , r ∈ R, the Fourier series of the distribution f (r) is Using this operator, we define the weighted Sobolev class as follows: For r >  and  ≤ p ≤ ∞, while the weighted Sobolev class BW r p,α,β is defined to be the unit ball of W r p,α,β . When p = , the norm · W r ,α,β is equivalent to the norm · W r ,α,β , and we can rewrite W r with the inner product f , g r := f , P  g, P  + f (r) , g (r) .

Main lemmas
We identify R m with the space m  , denote by x, y the Euclidean inner product of x, y ∈ R m , and write ·  instead of · m  . Consider in R m the standard Gaussian measure γ m , which is given by where G is any Borel subset in R m . Let  ≤ q ≤ ∞,  ≤ n < m, and δ ∈ [, ). The probabilistic linear (n, δ)-width of a linear mapping T : R m → l m q is defined by where G δ runs over all possible Borel subsets of R m with measure γ m (G δ ) ≤ δ, and T n runs over all linear operators from R m to l m q with rank at most n. Now, we introduce several lemmas which will be used in the proof of Theorem ..
Let λ n (t) be the Christoffel function and b j = λ n (ξ j ). Denote It is well known uniformly (see []) where the constants of equivalence depend only on α, β (see [] or []). The following lemma is well known as Gaussian quadrature formulae.
is exact for all polynomials of degree n -. Moreover, for any  ≤ p ≤ ∞, f ∈ n , we have An equivalence like (.) is generally called a Marcinkiewicz-Zygmund type inequality.
where S n is given in (.). Denote by the reproducing kernel of the Hilbert space L ,α,β ∩  k n= k- + P n . Then, for x ∈ [, ], By Lemma ., there exists a sequence of positive numbers and for x ∈ R  k+ , Let the operator T k : R  k+ − →  k+ be defined by where a := (a  , . . . , a  k+ ) ∈ R  k+ . It is shown in [] that for  ≤ q ≤ ∞, In what follows, we use the letters S k , R k , V k to denote u k × u k real diagonal matrixes as follows: and use the letter R - k to represent the inverse matrix of R k .
Proof Denote by K the set By the Riesz representation theorem and the Cauchy-Schwarz inequality, we have

Proofs of main results
Before Theorem . is proved, we establish the discretization theorems which give the reduction of the calculation of the probabilistic widths.
Theorem . Let  ≤ q ≤ ∞, σ ∈ (, ), and let the sequences of numbers {n k } and {σ k } be such that  ≤ n k ≤  k+ =: m k , ∞ k= n k ≤ n, σ k ∈ (, ), ∞ k= σ k ≤ σ . Then Proof For convenience, we write where γ m k is the standard Gaussian measure in R m k . Denote by L k a linear operator from R m k to R m k such that the rank of L k is at most n k and Then, for any f ∈ W r ,α,β , by (.)-(.), (.) and (.) we have where M (r  ,) k (x, y) is the r  -order partial derivative of M k (x, y) with respect to the variable x, r  ∈ R. Since the random vector f in W r ,α,β is a centered Gaussian random vector with a covariance operator C ν , the vector in R m k is a random vector with a centered Gaussian distribution γ in R m k , and its covariance matrix C γ is given by Since for any z = (z  , . . . , z m k ) ∈ R m k , Now we consider the subset of W r ,α,β where c  , c  are the positive constants given in (.), (.). Then by (.) we get Note that for any t > , the set {y ∈ R m k : V k y -L k y l m k q ≤ t} is convex symmetric. It then follows by Theorem .. in [] and (.), we have where λ is a centered Gaussian measure in R m k with covariance matrix c    -kρ I m k . Consider G = ∞ k= G k and the linear operator T n on W r ,α,β which is given by Then Thus, according to the definitions of G, T n , and L k , we obtain λ n,δ W r ,α,β , ν, L q,α,β = sup We may take b  >  sufficiently large so that N ≥ n. Let ϕ  be a C ∞ -function on R supported in [-, ], and be equal to  on [-/, /]. Let ϕ  be a nonnegative C ∞ -function on R supported in [-/, /], and be equal to  on [-/, /]. Define Clearly, It follows that for F a ∈ A n , a = (a  , . . . , a N ) ∈ R N , we have For a nonnegative integer ν = , , . . . , and F a ∈ A N , a = (a  , . . . , a N ) ∈ R N , it follows from the definition of -D α,β that Hence, for  ≤ q ≤ ∞ and F a = N j= a j ϕ j ∈ A N , It then follows by the Kolmogorov type inequality (see Theorem . in []) that For f ∈ L ,α,β and x ∈ [-, ], we define Clearly, the operator P N is the orthogonal projector from L ,α,β to A N , and if f ∈ W ρ ,α,β , then Q N (f )(x) = P N (f ρ )(x). Also, using the method in [], we can prove that P N is the bounded operator from L q,α,β to A N ∩ L q,μ for  ≤ q ≤ ∞, , and let N be given above. Then λ n,δ W r ,α,β , ν, L q,α,β n /-ρ-/q λ n,δ I N : where N n, N ≥ n and γ N is the standard Gaussian measure in R N .
Proof Let T n be a bounded linear operator on W r ,α,β with rank T n ≤ n such that where λ n,δ := λ n,δ (W r ,α,β , ν, L q,α,β ). Note that if A is a bounded linear operator from W r ,α,β to W r ,α,β and from H(ν) to H(ν), then the image measure λ of ν under A is also a centered Gaussian measure on W r ,α,β with covariance where C ν is the covariance of the measure ν, H(ν) = W ρ ,α,β is the Camera-Martin space of ν, and A * is the adjoint of A in H(ν) (see Theorem .. of []). Furthermore, if the operator A also satisfies By Theorem .. in [], we get that for any absolutely convex Borel set E of W r ,α,β there holds the inequality Applying (.) we assert that Then there exists a positive constant c  such that Note that, for any t > , the set {f ∈ W r ,α,β : f -T n f q,α,β ≤ t} is absolutely convex. It then follows that which leads to We see at once that L N J N (F a ) = F a for any F a ∈ A N . Set y = (y  , . . . , y N ) ∈ R N , where y j =  ϕ j ,α,β f , ϕ (ρ) j . Then y = J N Q N (f ). Thus by (.) and ϕ j ,α,β m -  , we obtain Combining (.) with (.), we conclude that for any f ∈ W r ,α,β , Remark that g k = ϕ k ϕ k ,α,β , k = , , . . . , N , is an orthonormal system in L ,α,β and g k ∈ H(v) = W ρ ,α,β . Then the random vector ( f , g (ρ)  , . . . , f , g (ρ) N ) = y in R N on the measurable space (W r ,α,β , ν) has the standard Gaussian distribution r N in R N . It then follows that where c  is a positive constant. Clearly, rank(J N P N T n L N ) ≤ n and r N (G) ≤ ν f ∈ W r ,α,β : f -T n f q,α,β > λ n,δ ≤ δ. Consequently, This completes the proof of Theorem .. Now, we are in a position to prove Theorem ..

Conclusions
In this paper, optimal estimates of the probabilistic linear (n, δ)-widths of the weighted Sobolev space W r ,α,β on [-, ] are established. This kind of estimates play an important role in the widths theory and have a wide range of applications in the approximation theory of functions, numerical solutions of differential and integral equations, and statistical estimates.