Open Access

Probabilistic linear widths of Sobolev space with Jacobi weights on \([-1,1]\)

Journal of Inequalities and Applications20172017:262

https://doi.org/10.1186/s13660-017-1540-7

Received: 6 March 2017

Accepted: 11 October 2017

Published: 23 October 2017

Abstract

Optimal asymptotic orders of the probabilistic linear \((n,\delta)\)-widths of \(\lambda_{n,\delta }(W^{r}_{2,\alpha,\beta }, \nu,L_{q,\alpha,\beta })\) of the weighted Sobolev space \(W_{2,{\alpha, \beta }}^{r}\) equipped with a Gaussian measure ν are established, where \(L_{q,\alpha,\beta }\), \(1\leq q\leq \infty \), denotes the \(L_{q}\) space on \([-1,1]\) with respect to the measure \((1-x)^{\alpha }(1+x)^{\beta }\), \(\alpha,\beta > -1/2\).

Keywords

probabilistic linear widthsJacobi weightsweighted Sobolev classesGaussian measure

MSC

41A4641A2528C2042C15

1 Introduction

This paper mainly focuses on the study of probabilistic linear \((n,\delta)\)-widths of a Sobolev space with Jacobi weights on the interval \([-1,1]\). This problem has been investigated only recently. For calculation of probabilistic linear \((n,\delta)\)-widths of the Sobolev spaces equipped with Gaussian measure, we refer to [15]. Let us recall some definitions.

Let K be a bounded subset of a normed linear space X with the norm \(\Vert \cdot \Vert _{X}\). The linear n-width of the set K in X is defined by
$$\lambda_{n}(K,X)= \inf_{L_{n}}\sup _{x\in K} \Vert x-L_{n}x\Vert _{X}, $$
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 \(\mathcal{B}\) which is the smallest σ-algebra containing all open subsets. Assume that ν is a probability measure defined on \(\mathcal{B}\). Let \(\delta \in [0,1)\). The probabilistic linear \((n,\delta)\)-width is defined by
$$\lambda_{n,\delta }(W,\nu, X)=\inf_{G_{\delta }} \lambda_{n}(W\backslash G_{\delta },X), $$
where \(G_{\delta }\) runs through all possible ν-measurable subsets of W with measure \(\nu (G_{\delta })\leq \delta \). Compared with the classical case analysis (see [2] or [6]), 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 [7], Wang has obtained the asymptotic orders of probabilistic linear \((n,\delta)\)-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 \((n,\delta)\)-widths on the interval \([-1,1]\) with Jacobi weights and determines the asymptotic orders of the probabilistic linear \((n,\delta)\)-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.

2 Main results

Consider the Jacobi weights
$$w_{\alpha,\beta }(x):=(1-x)^{\alpha }(1+x)^{\beta }, \quad \alpha, \beta > -1/2. $$
Denote by \(L_{p,\alpha,\beta }\equiv L_{p}(w_{\alpha,\beta })\), \(1 \le p<\infty \), the space of measurable functions defined on \([-1,1]\) with the finite norm
$$\Vert f\Vert _{p,\alpha,\beta }:= \biggl(\int_{-1}^{1}\bigl\vert f(x)\bigr\vert ^{p} w_{\alpha, \beta }(x)\,dx \biggr)^{1/p}, \quad 1\le p< \infty, $$
and for \(p=\infty \) we assume that \(L_{\infty,\alpha,\beta }\) is replaced by the space \(C[-1,1]\) of continuous functions on \([-1,1]\) with the uniform norm. Let \(\Pi_{n}\) be the space of all polynomials of degree at most n. Denote by \(\mathbb{P}_{n}\) the space of all polynomials of degree n which are orthogonal to polynomials of low degree in \(L_{2}(w_{\alpha,\beta })\). It is well known that the classical Jacobi polynomials \(\{P_{n}^{(\alpha,\beta)}\}_{n=0}^{\infty }\) form an orthogonal basis for \(L_{2,\alpha,\beta }:=L_{2}([-1,1],w_{\alpha, \beta })\) and are normalized by \(P_{n}^{(\alpha,\beta)}(1)= \bigl({\scriptsize\begin{matrix}{}n+\alpha \cr n\end{matrix}}\bigr)\) (see [8]). In particular,
$$\int_{-1}^{1} P_{n}^{(\alpha,\beta)}(x)P_{n}^{(\alpha,\beta)}(y)w _{\alpha,\beta }(x)\,dx=\delta_{n,m}h_{n}{(\alpha,\beta)}, $$
where
$$h_{n}{(\alpha,\beta)}=\frac{\Gamma (\alpha +\beta +2)}{\Gamma (\alpha +1)\Gamma (\beta +1)} \frac{\Gamma (n+\alpha +1)\Gamma (n+ \beta +1)}{(2n+\alpha +\beta +1)\Gamma (n+1)\Gamma (n+\alpha +\beta +1)} \sim n^{-1} $$
with constants of equivalence depending only on α and β. Then the normalized Jacobi polynomials \(P_{n}(x)\), defined by
$$P_{n}(x)=\bigl(h_{n}^{(\alpha,\beta)}\bigr)^{-1/2}P_{n}^{(\alpha,\beta)}(x),\quad n=0,1,\ldots, $$
form an orthonormal basis for \(L_{2,\alpha,\beta }\), where the inner product is defined by
$$\langle f,g\rangle:= \int_{-1}^{1} f(x) \overline{g(x)}w_{\alpha, \beta }(x)\,dx. $$
Denote by \(S_{n}\) the orthogonal projector of \(L_{2}(w_{\alpha, \beta })\) onto \(\Pi_{n}\) in \(L_{2}(w_{\alpha,\beta })\), which is called the Fourier partial summation operator. Consequently, for any \(f\in L_{2}(W_{\alpha,\beta })\),
$$ f=\sum_{l=0}^{\infty }\langle f,P_{l}\rangle P_{l} ,\qquad S_{n}f:=\sum _{l=0}^{n} \langle f,P_{l}\rangle P_{l} . $$
(2.1)
It is well known that (see Proposition 1.4.15 in [9]) \(P_{n}^{(\alpha,\beta)}\) is just the eigenfunction corresponding to the eigenvalues \(-n(n+\alpha +\beta +1)\) of the second-order differential operator
$$D_{\alpha,\beta }:= \bigl(1-x^{2}\bigr)D^{2}-\bigl(\alpha -\beta +(\alpha +\beta +2)x\bigr)D, $$
which means that
$$D_{\alpha,\beta }P_{n}^{(\alpha,\beta)}(x)=-n(n+\alpha +\beta +1)P _{n}^{(\alpha,\beta)}(x). $$
Given \(r>0\), we define the fractional power \((-D_{\alpha,\beta })^{r/2}\) of the operator \(-D_{\alpha,\beta }\) on f by
$$(-D_{\alpha,\beta })^{r/2} (f)= \sum_{k=0}^{\infty } \bigl(k(k+\alpha + \beta +1)\bigr)^{r/2} \langle f,P_{k} \rangle P_{k}, $$
in the sense of distribution. We call \(f^{(r)}:=(-D_{\alpha,\beta })^{r/2}\) the rth order derivative of the distribution f. It then follows that for \(f\in L_{2,\alpha,\beta } \), \(r\in R\), the Fourier series of the distribution \(f^{(r)}\) is
$$f^{(r)}=\sum_{k=1}^{\infty } \bigl(k(k+\alpha +\beta +1)\bigr)^{r/2}\langle f,P _{k}\rangle P_{k}. $$
Using this operator, we define the weighted Sobolev class as follows: For \(r>0\) and \(1\le p\le \infty \),
$$W_{p,\alpha,\beta }^{r}\bigl([-1,1]\bigr)\equiv W_{p,{\alpha,\beta }}^{r}:= \bigl\{ f\in L_{p,\alpha,\beta } : \Vert f\Vert _{W_{p,\alpha,\beta }^{r}}:= \Vert f \Vert _{p,\alpha,\beta }+\bigl\Vert (-D_{\alpha,\beta })^{\frac{r}{2}}(f)\bigr\Vert _{p,\alpha,\beta }< \infty \bigr\} , $$
while the weighted Sobolev class \(BW_{p,{\alpha,\beta }}^{r}\) is defined to be the unit ball of \(W_{p,{\alpha,\beta }}^{r}\). When \(p=2\), the norm \(\Vert \cdot \Vert _{ W_{2,\alpha,\beta }^{r}}\) is equivalent to the norm \(\Vert \cdot \Vert _{\overline{W}_{2,\alpha,\beta }^{r}}\), and we can rewrite \(W_{2,\alpha,\beta }^{r}\) as
$$\begin{aligned} W_{2,\alpha,\beta }^{r} =&\overline{W}_{2,\alpha,\beta }^{r} \\ :=& \Biggl\{ f(x) =\sum_{l=0}^{\infty }\langle f,P_{n}\rangle P_{n}(x): \Vert f\Vert _{\overline{W}_{2,\alpha,\beta }^{r}}^{2}:=\langle f,P_{0}\rangle ^{2}+\bigl\langle f^{(r)},f^{(r)}\bigr\rangle \\ &= \langle f,P_{0}\rangle^{2}+\sum _{k=1}^{\infty } \bigl(k(k+\alpha + \beta +1) \bigr)^{r}\langle f,P_{k}\rangle^{2}< \infty \Biggr\} \end{aligned}$$
with the inner product
$$\langle f,g\rangle_{r}:=\langle f,P_{0}\rangle \langle g,P_{0}\rangle +\bigl\langle f^{(r)},g^{(r)}\bigr\rangle . $$
Obviously, \(\overline{W}_{2,\alpha,\beta }^{r}\) is a Hilbert space. We equip \(\overline{W}_{2,\alpha,\beta }^{r}=W_{2,\alpha,\beta }^{r} \) with a Gaussian measure ν whose mean is zero and whose correlation operator \(C_{\nu }\) has eigenfunctions \(P_{l}(x)\), \(l=0,1,2,\dots \), and eigenvalues
$$\lambda_{0}=1,\qquad \lambda_{l}=\bigl(l(l+\alpha +\beta +1) \bigr)^{-s/2} ,\quad l=1,2, \dots, s>1, $$
that is,
$$C_{\nu }P_{0}=P_{0},\qquad C_{\nu }P_{l}= \lambda_{l} P_{l}, \quad l=1,2, \dots. $$
Then (see [10], pp.48-49),
$$\langle C_{\nu }f,g\rangle_{r}= \int_{\overline{W}_{2,{\alpha,\beta }}^{r}} \langle f,h\rangle_{r} \langle g,h \rangle_{r} \nu (dh). $$
By Theorem 2.3.1 of [10] the Cameron-Martin space \(H(\nu)\) of the Gaussian measure ν is \(\overline{W}_{2,{\alpha,\beta }}^{r+s/2}\), i.e.,
$$H(\nu)= \overline{W}_{2,{\alpha,\beta }}^{r+s/2}. $$

See [10] and [11] for more information about the Gaussian measure on Banach spaces.

Throughout the paper, \(A(n,\delta)\asymp B(n,\delta)\) means \(A(n,\delta)\ll B(n,\delta)\) and \(A(n,\delta)\gg B(n,\delta)\), \(A(n,\delta)\ll B(n,\delta)\) means that there exists a positive constant c independent of n and δ such that \(A(n,\delta) \le cB(n,\delta)\). If \(1\le q\le \infty\), \(r>(2+2\min \{0,\max \{ \alpha,\beta \}\})(1/p-1/q)_{+}\), the space \(W_{p,\alpha,\beta } ^{r}\) can be continuously embedded into the space \(L_{q,{\alpha, \beta }}\) (see Lemma 2.3 in [12]).

Set \(\rho =r+\frac{s}{2}\). The main result of this paper can be formulated as follows.

Theorem 2.1

Let \(1\le q \le \infty\), \(\delta \in (0,1/2] \), and let \(\rho >1/2+(2\max \{\alpha,\beta \}+1)(1/2+1/q)_{+}\). Then
$$ \lambda_{n,\delta }\bigl(W_{2,\alpha,\beta }^{r},\nu,L_{q,\alpha,\beta }\bigr)\asymp \textstyle\begin{cases} n^{1/2-\rho }(1+n^{-\min \{1/2,1/q\}})(\ln (\frac{1}{\delta }))^{ \frac{1}{2}}, & 1\leq q< \infty, \\ n^{1/2-\rho }(\ln (\frac{n}{\delta }))^{\frac{1}{2}}, & q=\infty. \end{cases} $$
(2.2)

For the proof of Theorem 2.1, the discretization technique is used (see [1, 4, 13, 14]). Since the known results of the probabilistic linear widths of the identity matrix on \(\mathbb{R}^{m}\) are inappropriate here, the probabilistic linear widths of diagonal matrixes on \(\mathbb{R}^{m}\) are adopted for the proof of the upper estimates.

3 Main lemmas

Let \(\ell_{q}^{m}\) (\(1\le q\le \infty\)) denote the space \(\mathbb{R}^{m}\) equipped with the \(\ell_{q}^{m}\)-norm defined by
$$\Vert x\Vert _{\ell_{q}^{m}}:= \textstyle\begin{cases} (\sum_{i=1}^{m} \vert x_{i}\vert ^{q} )^{\frac{1}{q}}, &1\le q< \infty, \\ \max_{1\le i\le m} \vert x_{i}\vert , &q=\infty. \end{cases} $$
We identify \(\mathbb{R}^{m}\) with the space \(\ell_{2}^{m}\), denote by \(\langle x,y\rangle \) the Euclidean inner product of \(x,y\in \mathbb{R} ^{m}\), and write \(\Vert \cdot \Vert _{2}\) instead of \(\Vert \cdot \Vert _{\ell_{2} ^{m}}\).
Consider in \(\mathbb{R}^{m}\) the standard Gaussian measure \(\gamma_{m}\), which is given by
$$\gamma_{m}(G)=(2\pi)^{-m/2} \int_{G} \exp^{\frac{-\Vert x\Vert ^{2}}{2}}\,dx, $$
where G is any Borel subset in \(\mathbb{R}^{m}\). Let \(1\le q\le \infty \), \(1\le n< m\), and \(\delta \in [0,1)\). The probabilistic linear \((n,\delta)\)-width of a linear mapping \(T:\mathbb{R}^{m}\rightarrow l ^{m}_{q}\) is defined by
$$\lambda_{n,\delta }\bigl(T:\mathbb{R}^{m}\rightarrow l^{m}_{q},\gamma_{m}\bigr)= \inf _{G_{\delta }}\inf_{T_{n}}\sup_{\mathbb{R}^{m}\setminus G_{\delta }} \Vert Tx-T_{n}x\Vert _{l_{q}^{m}}, $$
where \(G_{\delta }\) runs over all possible Borel subsets of \(\mathbb{R}^{m}\) with measure \(\gamma_{m}(G_{\delta })\leq \delta \), and \(T_{n}\) runs over all linear operators from \(\mathbb{R}^{m}\) to \(l_{q}^{m}\) with rank at most n.
Throughout the paper, D denotes the \(m\times m\) real diagonal matrix \(\operatorname{diag}(d_{1},\ldots,d_{m})\) with \(d_{1}\geq d_{2}\ge \cdots \ge d_{m}>0\), \(D_{n}\) denotes the \(m\times m\) real diagonal matrix \(\operatorname{diag}(d_{1},\ldots,d_{n},0,\ldots,0)\) with \(1\le n\le m\), and \(I_{m}\) denotes the \(m\times m\) identity matrix. Moreover, \(\{e_{1},\ldots,e_{m}\}\) denotes the standard orthonormal basis in \(\mathbb{R}^{m}\):
$$e_{1}=(1,0,\ldots,0),\qquad \ldots,\qquad e_{m}=(0,\ldots,0,1). $$

Now, we introduce several lemmas which will be used in the proof of Theorem 2.1.

Lemma 3.1

  1. (1)
    (See [1]) If \(1\le q\le 2\), \(m\ge 2n\), \(\delta \in (0,1/2]\), then
    $$ \lambda_{n,\delta }\bigl(I_{m}:\mathbb{R}^{m} \rightarrow l^{m}_{q},\gamma_{m}\bigr) \asymp m^{1/q}+m^{1/q-1/2}\sqrt{\ln (1/\delta)}. $$
    (3.1)
     
  2. (2)
    (See [4]) If \(2\le q<\infty \), \(m\ge 2n\), \(\delta \in (0,1/2]\), then
    $$ \lambda_{n,\delta }\bigl(I_{m}:\mathbb{R}^{m} \rightarrow l^{m}_{q},\gamma_{m}\bigr) \asymp m^{1/q}+\sqrt{\ln (1/\delta)}. $$
    (3.2)
     
  3. (3)
    (See [5]) If \(q=\infty \), \(m\ge 2n\), \(\delta \in (0,1/2]\), then
    $$ \lambda_{n,\delta }\bigl(I_{m}:\mathbb{R}^{m} \rightarrow l^{m}_{q},\gamma_{m}\bigr) \asymp \sqrt{\ln \bigl((m-n)/\delta }\bigr)\asymp \sqrt{\ln m+\ln (1/\delta)}. $$
    (3.3)
     

Lemma 3.2

(See [7])

Assume that
$$\sum_{i=1}^{m}d_{i}^{\beta } \le C(m,\beta)\quad \textit{for some }\beta >0. $$
Then, for \(2\le q\le \infty \), \(m\ge 2n\), \(\delta \in (0,1/2]\), we have
$$ \lambda_{n,\delta }\bigl(D:\mathbb{R}^{m}\rightarrow l^{m}_{q},\gamma_{m}\bigr) \ll \biggl(\frac{C(m,\beta)}{n+1} \biggr)^{\frac{1}{\beta }} \textstyle\begin{cases} (m^{1/q}+\sqrt{\ln (1/\delta)}, &2\le q< \infty, \\ \sqrt{\ln m+\ln (1/\delta)}, &q=\infty. \end{cases} $$
(3.4)
Let \(\xi_{j}=\cos \theta_{j}\), \(1\le j\le 2n\), denote the zeros of the Jacobi polynomial \(P_{2n}^{(\alpha,\beta)}(t)\), ordered so that
$$0=:\theta_{0}< \theta_{1}< \cdots < \theta_{2n}< \theta_{2n+1}:=\pi. $$
Let \(\lambda_{2n}(t)\) be the Christoffel function and \(b_{j}=\lambda _{2n}(\xi_{j})\). Denote
$$W(n;\xi_{j})=\bigl(1-x+n^{-2}\bigr)^{\alpha +\frac{1}{2}} \bigl(1-x+n^{-2}\bigr)^{\beta + \frac{1}{2}}. $$
It is well known uniformly (see [15])
$$\theta_{j+1}-\theta_{j}\asymp n^{-1},\qquad \theta_{j}\asymp jn^{-1} \quad (1 \le j\le 2n), $$
and also
$$b_{j}\asymp n^{-1}w_{\alpha,\beta }(\xi_{j}) \bigl(1-\xi_{j}^{2}\bigr)^{1/2} \asymp n^{-1}W(n;\xi_{j}), $$
where the constants of equivalence depend only on α, β (see [16] or [17]).

The following lemma is well known as Gaussian quadrature formulae.

Lemma 3.3

(See [8])

For each \(n\ge 1\), the quadrature
$$ \int_{-1}^{1} f(x)w_{\alpha,\beta }(x)\,dx\asymp \sum _{j=1}^{2n}b _{j} f(\xi_{j}) $$
(3.5)
is exact for all polynomials of degree \(4n-1\). Moreover, for any \(1\le p\le \infty\), \(f\in \Pi_{n}\), we have
$$ \Vert f\Vert _{p,\alpha,\beta }\asymp \Biggl(\sum _{j=1}^{2n}b_{j} \bigl\vert f(\xi_{j})\bigr\vert ^{p} \Biggr)^{1/p}. $$
(3.6)

An equivalence like (3.6) is generally called a Marcinkiewicz-Zygmund type inequality.

Lemma 3.4

(See [12], Lemma 2.7)

Let \(\alpha,\beta >-1/2\), \(\sigma \in (0,\frac{1}{2\max \{\alpha,\beta \}+1})\) and let \(b_{j}\), \(1\le j\le n\), be defined as in Lemma  3.3. Then
$$ \sum_{j=1}^{n} b_{j}^{-\sigma } \ll n^{1+\sigma }. $$
(3.7)
Let
$$ L_{n}(x,y):=\sum_{j=0}^{\infty }\eta \biggl(\frac{j}{n}\biggr)P_{j}(x)P_{j}(y),\quad x,y\in [-1,1], $$
(3.8)
where \(\eta \in C^{\infty }(R)\) is a nonnegative \(C^{\infty }\)-function on \([0,\infty)\) supported in \([0,2]\) with the properties that \(\eta (t)=1\) for \(0\leq t\leq 1\) and \(\eta (t)>0\) for \(t\in [0,2)\). For any \(f\in L_{2,\alpha,\beta }\), we define
$$ \delta_{1}(f)=S_{2}(f),\qquad \delta_{k}(f)=S_{2^{k}}(f)-S_{2^{k-1}}(f) \quad \mbox{for } k=2,3\ldots, $$
(3.9)
where \(S_{n}\) is given in (2.1). Denote by
$$ M_{k}(x,y)=\sum_{l=2^{k-1}+1}^{2^{k}}P_{l}(x)P_{l}(y) $$
(3.10)
the reproducing kernel of the Hilbert space \(L_{2,\alpha,\beta } \cap \bigoplus_{n=2^{k-1}+1}^{2^{k}}\mathbb{P}_{n}\). Then, for \(x\in [0,1]\),
$$\delta_{k}(f) (x)=\sum_{l=2^{k-1}+1}^{2^{k}} \int_{-1}^{1}f(x)P_{l}(x)P _{l}(y)w_{\alpha,\beta }(y)\,dx=\bigl\langle f,M_{k}(\cdot,x)\bigr\rangle . $$
For \(f\in \bigoplus_{n=2^{k-1}+1}^{2^{k}}\mathbb{P}_{n}\),
$$f(x)=\delta_{k}(f) (x)=\bigl\langle f,M_{k}(\cdot,x)\bigr\rangle . $$
By Lemma 3.3, there exists a sequence of positive numbers \(w_{i}=b _{i}\asymp n^{-1}W_{\alpha,\beta }(n;\xi_{i})\), \(1\le i\le 2^{k+1}\), for which the following quadrature formula holds for all \(f \in \Pi_{2^{k+3}-1}\):
$$ \int_{-1}^{1} f(t)W_{\alpha,\beta }(t)\,dt=\sum _{i=1}^{2^{k+1}}w_{i} f(\xi_{i}). $$
(3.11)
Moreover, for any \(1\le p\le \infty\), \(f\in \Pi_{2^{k}}\), we have
$$\Vert f\Vert _{p,\alpha,\beta }\asymp \Biggl(\sum _{i=1}^{2^{k+1}}w_{i} \bigl\vert f(\xi_{i})\bigr\vert ^{p} \Biggr)^{1/p}=\bigl\Vert U_{n}(f)\bigr\Vert _{\ell_{p,w}^{2^{k+1}}}, $$
where \(w=(w_{1},\dots,w_{2^{k+1}})\), \(U_{k}:\Pi_{2^{k}}\longmapsto \mathbb{R}^{2^{k+1}}\) is defined by
$$ U_{k}(f)=\bigl(f(\xi_{1}),\dots,f(\xi_{2^{k+1}}) \bigr), $$
(3.12)
and for \(x\in \mathbb{R}^{2^{k+1}}\),
$$\Vert x\Vert _{\ell_{p,w}^{2^{k+1}}}:=\textstyle\begin{cases} (\sum_{i=1}^{2^{k+1}} \vert x_{i}\vert ^{p}w_{i} )^{\frac{1}{p}}, &1\le p< \infty, \\ \max_{1\le i\le {2^{k+1}}} \vert x_{i}\vert , &p=\infty. \end{cases} $$
Let the operator \(T_{k}:\mathbb{R}^{2^{k+1}}\longmapsto \Pi_{2^{k+1}}\) be defined by
$$ T_{k}a(x):=\sum_{i=1}^{2^{k+1}}a_{i}w_{i}L_{2^{k+1}}(x, \xi_{i}), $$
(3.13)
where \(a:=(a_{1},\dots,a_{2^{k+1}})\in \mathbb{R}^{2^{k+1}}\). It is shown in [12] that for \(1\le q\le \infty \),
$$ \Vert T_{k}a\Vert _{q,\alpha,\beta }\ll \Vert v\Vert _{\ell_{q,w}^{2^{k+1}}}. $$
(3.14)
For \(f\in \Pi_{2^{k+1}}\), we have
$$f(x)= \int_{-1}^{1}f(y)L_{2^{k+1}}(x,y)w_{\alpha,\beta }(x,y)\,dy= \sum_{i=1}^{2^{k+1}}w_{i}f(\xi_{i})L_{2^{k+1}}(x,\xi_{i})=T_{k}U_{k}(f) (x). $$
In what follows, we use the letters \(S_{k}\), \(R_{k}\), \(V_{k}\) to denote \(u_{k}\times u_{k}\) real diagonal matrixes as follows:
$$ \begin{aligned} &S_{k}=\operatorname{diag}\bigl(w_{1}^{\frac{1}{2}},\ldots,w_{2^{k+1}}^{\frac{1}{2}}\bigr), \\ & R_{k}=\operatorname{diag}\bigl(w_{1}^{\frac{1}{q}},\ldots,w_{2^{k+1}}^{\frac{1}{q}} \bigr), \\ &V_{k}=\operatorname{diag}\bigl(w_{1}^{-\frac{1}{2}+\frac{1}{q}},\ldots,w_{2^{k+1}}^{- \frac{1}{2}+\frac{1}{q}}\bigr), \end{aligned}$$
(3.15)
and use the letter \(R_{k}^{-1}\) to represent the inverse matrix of \(R_{k}\).

Lemma 3.5

For any \(z=(z_{1},\ldots,z_{2^{k+1}})\in \mathbb{R}^{2^{k+1}}\), we have
$$ \Biggl\Vert \sum_{j=1}^{2^{k+1}}w_{j}^{\frac{1}{2}}z_{j}M_{k}(\cdot,\xi_{j})\Biggr\Vert _{2,\alpha,\beta }\ll \Vert z\Vert _{l_{2}^{2^{k+1}}}, $$
(3.16)
where \(M_{k}(x,y)\) is given in (3.10), and \((\xi_{1},\ldots,\xi_{2^{k+1}})\) is defined as above.

Proof

Denote by K the set
$$\Biggl\{ g\in \bigoplus^{2^{k}}_{j=2^{k-1}-1} \mathbb{P}_{j} : \Vert g\Vert _{2,\alpha, \beta } \leq 1 \Biggr\} . $$
Since
$$\sum^{2^{k+1}}_{j=1} w^{1/2}_{j} z_{j} M_{k}(\cdot,\xi_{j}) \in L_{2, \alpha,\beta } \cap \Biggl(\bigoplus^{2^{k}}_{j=2^{k-1}-1} \mathbb{P}_{j}\Biggr). $$
By the Riesz representation theorem and the Cauchy-Schwarz inequality, we have
$$\begin{aligned} \Biggl\Vert \sum^{2^{k+1}}_{j=1} w^{1/2}_{j} z_{j}M_{k}(\cdot, \xi_{j})\Biggr\Vert _{2,\alpha,\beta } =& \sup_{g \in K} \Biggl\vert \Biggl\langle \sum^{2^{k+1}}_{j=1} w^{1/2}_{j} z_{j} M_{k}(\cdot , \xi_{j}),g \Biggr\rangle \Biggr\vert \\ =& \sup_{g \in K} \Biggl\vert \sum^{2^{k+1}}_{j=1} w^{1/2}_{j} z_{j} g(\xi_{j}) \Biggr\vert \\ \leq & \sup_{g \in K} \Biggl(\sum^{2^{k+1}}_{j=1} \vert z_{j}\vert ^{2}\Biggr)^{1/2} \Biggl(\sum^{2^{k+1}}_{j=1} w_{j}\bigl\vert g(\xi_{j})\bigr\vert ^{2}\Biggr)^{1/2} \\ \ll & \sup_{g \in K} \Biggl(\sum^{2^{k+1}}_{j=1} \vert z_{j}\vert ^{2}\Biggr)^{1/2} \Vert g \Vert _{2,\alpha,\beta } \\ \leq& \Vert z \Vert _{l^{2^{k+1}}_{2}}. \end{aligned}$$
 □

4 Proofs of main results

Before Theorem 2.1 is proved, we establish the discretization theorems which give the reduction of the calculation of the probabilistic widths.

Theorem 4.1

Let \(1\leq q\leq \infty\), \(\sigma \in (0,1)\), and let the sequences of numbers \(\{n_{k}\}\) and \(\{\sigma_{k}\}\) be such that \(0 \leq n_{k} \leq 2^{k+1}=:m_{k}\), \(\sum^{\infty }_{k=1} n_{k} \leq n\), \(\sigma_{k}\in (0,1)\), \(\sum^{\infty }_{k=1} \sigma_{k} \leq \sigma \). Then
$$ \lambda_{n,\sigma }\bigl(W^{r}_{2,\alpha,\beta },\nu,L_{q,\alpha,\beta }\bigr) \leq \sum^{\infty }_{k=1}2^{-k\rho } \lambda_{n_{k},\sigma_{k}}\bigl(V _{k} :\mathbb{R}^{m_{k}} \rightarrow l^{m_{k}}_{q},\gamma_{m_{k}}\bigr). $$
(4.1)

Proof

For convenience, we write
$$\lambda_{n_{k},\sigma_{k}}:=\lambda_{n_{k},\sigma_{k}}\bigl(V_{k} : \mathbb{R} ^{m_{k}} \rightarrow l^{m_{k}}_{q}, \gamma_{m_{k}}\bigr), $$
where \(\gamma_{m_{k}}\) is the standard Gaussian measure in \(\mathbb{R} ^{m_{k}}\). Denote by \(L_{k}\) a linear operator from \(\mathbb{R}^{m_{k}}\) to \(\mathbb{R}^{m_{k}}\) such that the rank of \(L_{k}\) is at most \(n_{k}\) and
$$\gamma_{m_{k}}\bigl(\bigl\{ y\in \mathbb{R}^{m_{k}} \vert \Vert V_{k} y-L_{k}y\Vert >2 \lambda_{n_{k},\sigma_{k}} \bigr\} \bigr) \leq \sigma_{k}. $$
Then, for any \(f \in W^{r}_{2,\alpha,\beta }\), by (3.8)-(3.10), (3.14) and (3.15) we have
$$\begin{aligned} \bigl\Vert \delta_{k}(f)-T_{k}R^{-1}_{k}L_{k}S_{k}U_{k} \delta_{k}(f) \bigr\Vert _{q, \alpha,\beta } =&\bigl\Vert T_{k}U_{k}\delta_{k}(f)-T_{k}R^{-1}_{k}L_{k}S_{k}U_{k} \delta_{k}(f)\bigr\Vert _{q,\alpha,\beta } \\ \leq& \bigl\Vert U_{k}\delta_{k}(f)-R^{-1}_{k}L_{k}S_{k}U_{k} \delta_{k}(f) \bigr\Vert _{l^{m_{k}}_{q,w}} \\ =&\bigl\Vert V_{k}S_{k}U_{k} \delta_{k}(f)-L_{k}S_{k}U_{k} \delta_{k}(f) \bigr\Vert _{l ^{m_{k}}_{q}}. \end{aligned}$$
(4.2)
Let \(y=S_{k}U_{k}\delta_{k}(f)=(w^{\frac{1}{2}}_{1} \delta_{k}(f)(\xi _{1}),\ldots,w^{\frac{1}{2}}_{m_{k}} \delta_{k}(f)(\xi_{m_{k}})) \in \mathbb{R}_{m_{k}}\), for \(x\in [-1,-1]\),
$$\delta_{k}(f) (x)=\bigl\langle f,M_{k}(\cdot,x)\bigr\rangle = \bigl\langle f^{(-r)},M ^{(-r,0)}_{k}(\cdot,x) \bigr\rangle _{r} = \bigl\langle f,M^{(-2r,0)}_{k}(\cdot,x)\bigr\rangle _{r}, $$
where \(M^{(r_{1},0)}_{k}(x,y)\) is the \(r_{1}\)-order partial derivative of \(M_{k}(x,y)\) with respect to the variable \(x,r_{1} \in \mathbb{R}\). Since the random vector f in \(W^{r}_{2,\alpha,\beta }\) is a centered Gaussian random vector with a covariance operator \(C_{\nu }\), the vector
$$y=S_{k}U_{k}\delta_{k}(f)=\bigl(\bigl\langle f,w^{\frac{1}{2}}_{1} M^{(-2r,0)} _{k}(\cdot, \xi_{1})\bigr\rangle _{r},\ldots,w^{\frac{1}{2}}_{m_{k}} M^{(-2r,0)} _{k}(\cdot,\xi_{m_{k}})\rangle_{r} \bigr) $$
in \(\mathbb{R}^{m_{k}}\) is a random vector with a centered Gaussian distribution γ in \(\mathbb{R}^{m_{k}}\), and its covariance matrix \(C_{\gamma }\) is given by
$$C_{\gamma }= \bigl(\bigl\langle C_{\nu }\bigl(w^{\frac{1}{2}}_{i} M^{(-2r,0)}_{k} (\cdot,\xi_{i}) \bigr),w^{\frac{1}{2}}_{j} M^{(-2r,0)}_{k} (\cdot, \xi_{j}) \bigr\rangle _{r} \bigr)^{m_{k}}_{i,j=1}. $$
Since for any \(z = (z_{1},\ldots,z_{m_{k}}) \in \mathbb{R}^{m_{k}}\),
$$\sum^{m_{k}}_{j=1}w_{j}^{\frac{1}{2}} z_{j} M_{k}(\cdot,\xi_{j}) \in \bigoplus _{j=2^{k-1}+1}^{2^{k}}\mathbb{P}_{j}, $$
and
$$\begin{aligned} \bigl\langle C_{\nu }\bigl(w^{\frac{1}{2}}_{i} M^{(-2r,0)}_{k} (\cdot,\xi_{i})\bigr),w ^{\frac{1}{2}}_{j} M^{(-2r,0)}_{k} (\cdot, \xi_{j})\bigr\rangle _{r} =& \bigl\langle w^{\frac{1}{2}}_{i} M^{(-2r-s,0)}_{k} (\cdot, \xi_{i}),w^{ \frac{1}{2}}_{j} M^{(-2r,0)}_{k} (\cdot,\xi_{j})\bigr\rangle _{r} \\ =& \bigl\langle w^{\frac{1}{2}}_{i} M^{(-\rho,0)}_{k} (\cdot,\xi_{i}),w ^{\frac{1}{2}}_{j} M^{(-\rho,0)}_{k} (\cdot,\xi_{j})\bigr\rangle , \end{aligned}$$
by Lemma 3.5 we get
$$\begin{aligned} \int_{\mathbb{R}^{m_{k}}}(y,z)^{2} \gamma (dy) =&z C_{\gamma }z^{T} = \sum^{m_{k}}_{i,j=1} z_{i} z_{j} \bigl\langle w^{\frac{1}{2}}_{i} M^{(- \rho,0)}_{k} (\cdot,\xi_{i}),w^{\frac{1}{2}}_{j} M^{(-\rho,0)}_{k} (\cdot,\xi_{j})\bigr\rangle \\ =& \Biggl\langle \sum^{m_{k}}_{j=1} w^{\frac{1}{2}}_{j} z_{j} M^{(-\rho,0)} _{k} (\cdot,\xi_{j}),\sum^{m_{k}}_{j=1} w^{\frac{1}{2}}_{j} z_{j} M ^{(-\rho,0)}_{k} (\cdot,\xi_{j}) \Biggr\rangle \\ =& \Biggl\Vert \sum^{m_{k}}_{j=1} w^{\frac{1}{2}}_{j} z_{j} M^{(-\rho,0)}_{k}(\cdot,\xi_{j})\Biggr\Vert ^{2}_{2} \asymp 2^{-2k\rho } \Biggl\Vert \sum^{m_{k}}_{j=1}w^{\frac{1}{2}}_{j} z_{j} M_{k} (\cdot,\xi_{j})\Biggr\Vert ^{2}_{2} \\ \ll& 2^{-2k\rho }\Vert z\Vert _{l_{2}^{m_{k}}} = 2^{-2k\rho } \int_{\mathbb{R}^{m_{k}}}(y,z)^{2} \gamma_{m_{k}}(dy). \end{aligned}$$
(4.3)
Now we consider the subset of \(W^{r}_{2,\alpha,\beta }\)
$$G_{k} :=\bigl\{ f \in W^{r}_{2,\alpha,\beta } \vert \bigl\Vert \delta_{k}(f)-T_{k}R^{-1}_{k}L_{k}S_{k}U_{k} \delta_{k}(f)\bigr\Vert _{l_{q}^{m_{k}}} >2c_{1}c_{2} 2^{-k\rho } \lambda_{n_{k},\sigma_{k}} \bigr\} , $$
where \(c_{1}\), \(c_{2}\) are the positive constants given in (4.2), (4.3). Then by (4.2) we get
$$\begin{aligned} \nu (G_{k}) \leq &\nu \bigl(\bigl\{ f \in W^{r}_{2,\alpha,\beta } \vert \bigl\Vert V_{k}S_{k}U_{k} \delta_{k}(f)-L_{k}S_{k}U_{k} \delta_{k}(f)\bigr\Vert _{l_{q} ^{m_{k}}} >2c_{2} 2^{-k\rho } \lambda_{n_{k},\sigma_{k}} \bigr\} \bigr) \\ =&\gamma \bigl(\bigl\{ y \in \mathbb{R}^{m_{k}} \vert \Vert V_{k} y-L_{k} y\Vert _{l_{q}^{m_{k}}} >2c_{2} 2^{-k\rho } \lambda_{n_{k},\sigma_{k}} \bigr\} \bigr). \end{aligned}$$
Note that for any \(t>0\), the set \(\{ y \in \mathbb{R}^{m_{k}} : \Vert V_{k} y-L_{k} y \Vert _{l_{q}^{m_{k}}} \leq t \}\) is convex symmetric. It then follows by Theorem 1.8.9 in [10] and (4.3), we have
$$\begin{aligned} \nu (G_{k}) \leq & \gamma \bigl(\bigl\{ y \in \mathbb{R}^{m_{k}} : \Vert V_{k} y-L_{k} y\Vert _{l_{q}^{m_{k}}} >2c_{2} 2^{-k\rho } \lambda_{n_{k},\sigma_{k}} \bigr\} \bigr) \\ \leq & \lambda \bigl(\bigl\{ y \in \mathbb{R}^{m_{k}} : \Vert V_{k} y-L_{k} y\Vert _{l_{q}^{m_{k}}} >2c_{2} 2^{-k\rho } \lambda_{n_{k},\sigma _{k}} \bigr\} \bigr) \\ \leq & \gamma_{m_{k}} \bigl(\bigl\{ y \in \mathbb{R}^{m_{k}} : \Vert V_{k}y-L_{k} y\Vert _{l_{q}^{m_{k}}} >2 \lambda_{n_{k},\sigma_{k}} \bigr\} \bigr) \leq \sigma_{k}, \end{aligned}$$
where λ is a centered Gaussian measure in \(\mathbb{R}^{m_{k}}\) with covariance matrix \(c_{2}^{2} 2^{-2k\rho } I_{m_{k}}\). Consider \(G=\bigcup^{\infty }_{k=1} G_{k}\) and the linear operator \(\widetilde{T}_{n} \) on \(W^{r}_{2,\alpha,\beta }\) which is given by
$$\widetilde{T}_{n} f = \sum^{\infty }_{k=1} T_{k}R^{-1}_{k}L_{k}S_{k}U _{k}\delta_{k}(f). $$
Then
$$\nu (G) = \nu \Biggl(\bigcup^{\infty }_{k=1} G_{k}\Biggr) \leq \sum^{\infty }_{k=1} \nu (G_{k}) \leq \sum^{\infty }_{k=1} \nu (\sigma_{k}) \leq \sigma, $$
and
$$\begin{aligned} \operatorname{rank}\widetilde{T}_{n} \leq& \sum^{\infty }_{k=1} \operatorname{rank}\bigl(T_{k}R^{-1}_{k}L _{k}S_{k}U_{k} \delta_{k}\bigr) \\ \leq& \sum^{\infty }_{k=1}n_{k} \leq n. \end{aligned}$$
Thus, according to the definitions of G, \(\widetilde{T}_{n}\), and \({L_{k}}\), we obtain
$$\begin{aligned} \lambda_{n,\delta }\bigl(W^{r}_{2,\alpha,\beta },\nu,L_{q,\alpha,\beta }\bigr) =& \sup_{f \in W^{r}_{2,\alpha,\beta } \backslash G} \Vert f- \widetilde{T}_{n}f \Vert _{q,\alpha,\beta } \\ \leq &\sup_{f \in W^{r}_{2,\alpha,\beta } \backslash G} \sum^{ \infty }_{k=1} \bigl\Vert \delta_{k}(f)-T_{k}R^{-1}_{k}L_{k}S_{k}U_{k} \delta_{k}(f) \bigr\Vert _{q,\alpha,\beta } \\ \leq & \sum^{\infty }_{k=1} \sup _{f \in W^{r}_{2,\alpha,\beta } \backslash G} \bigl\Vert \delta_{k}(f)-T_{k}R^{-1}_{k}L_{k}S_{k}U_{k} \delta_{k}(f) \bigr\Vert _{q,\alpha,\beta } \\ \ll & \sum^{\infty }_{k=1} 2^{-k \rho } \lambda_{n_{k},\sigma_{k}}, \end{aligned}$$
which completes the proof of Theorem 4.1. □
Now we turn to the lower estimates. Assume that \(m \geq 6\) and \(b_{1}m \leq n \leq 2b_{1}m\) with \(b_{1}>0\) being independent of n and m. Set \(\{x_{j}\}^{N}_{j=1}\subset \{x\in [-1,1]: \vert x\vert \leq 2/3\}\) and \(x_{j+1}-x_{j}=3/m\), \(j=1,\ldots, N-1\). Then \(M\asymp N \) and
$$\bigl\{ x\in [-1,1] : \vert x-x_{j}\vert \leq 1/m \bigr\} \cap \bigl\{ x\in [-1,1]: \vert x-x_{i}\vert \leq 1/m\bigr\} = \emptyset, \quad \mbox{if }i \neq j. $$
We may take \(b_{1}>0\) sufficiently large so that \(N\geq 2n\). Let \(\varphi^{1}\) be a \(C^{\infty }\)-function on \(\mathbb{R}\) supported in \([-1,1]\), and be equal to 1 on \([-2/3,2/3]\). Let \(\varphi^{2}\) be a nonnegative \(C^{\infty }\)-function on \(\mathbb{R}\) supported in \([-1/2,1/2]\), and be equal to 1 on \([-1/4,1/4]\). Define
$$\varphi_{i}(x)= \varphi^{1}\bigl(m(x-x_{i}) \bigr) - c_{i}\varphi^{2}\bigl(m(x-x_{i}) \bigr), $$
for some \(c_{i}\) such that \(\int^{1}_{-1} \varphi_{i}(x) W_{\alpha, \beta }(x)\,dx =0\), \(i=1,\ldots,N\). Set
$$A_{N} : = \operatorname{span}\{\varphi_{1},\ldots,\varphi_{N} \} = \Biggl\{ F_{a}(x) =\sum^{N}_{j=1} a_{j} \varphi_{j}(x): a=(a_{1},\ldots,a_{N}) \in \mathbb{R} ^{N} \Biggr\} . $$
Clearly,
$$\begin{aligned}& \varphi_{j} \in W^{2}_{2,\alpha,\beta },\quad \operatorname{supp}\varphi_{j} \subset \bigl\{ x \in [-1,1] : \vert x-x_{j} \vert \leq 1/m \bigr\} \subset \bigl\{ x \in [-1,1] : \vert x\vert \leq 5/6 \bigr\} , \\& \Vert \varphi_{j}\Vert _{q,\alpha,\beta } \asymp \biggl(\int^{2/3}_{-2/3} \bigl\vert \varphi_{j}(x) \bigr\vert ^{q}\,dx \biggr)^{1/q} = \biggl(\int^{2/3}_{-2/3} \bigl\vert \varphi^{1} \bigl(m(x-x_{j})\bigr)-c_{j} \varphi^{2} \bigl(m(x-x_{j})\bigr)\bigr\vert ^{q}\,dx \biggr)^{1/q} \\& \hphantom{\Vert \varphi_{j}\Vert _{q,\alpha,\beta }} \asymp m^{-1/q},\quad 1\leq q \leq \infty, j=1,\ldots,N, \end{aligned}$$
and
$$\operatorname{supp}\varphi_{j} \cap \operatorname{supp}\varphi_{i} = \emptyset\quad (i\neq j). $$
It follows that for \(F_{a}\in A_{n}\), \(a=(a_{1},\ldots,a_{N}) \in \mathbb{R}^{N}\), we have
$$ \Vert F_{a}\Vert _{q,\alpha,\beta }\asymp \Biggl(m^{-1} \sum^{N}_{j=1} \vert a_{j} \vert ^{q}\Biggr)^{1/q} = m^{-1/q}\Vert a\Vert _{l^{N}_{q}}. $$
(4.4)
For a nonnegative integer \(\nu =0,1,\ldots \) , and \(F_{a} \in A_{N}\), \(a=(a_{1},\ldots,a_{N}) \in \mathbb{R}^{N} \), it follows from the definition of \(-D_{\alpha,\beta } \) that
$$\operatorname{supp}(-D_{\alpha,\beta })^{\nu }(\varphi_{j})\subset \bigl\{ x \in [-1,1] : \vert x-x_{j}\vert \leq 1/m \bigr\} $$
and
$$\bigl\Vert (D_{\alpha,\beta })^{\nu }(\varphi_{j})\bigr\Vert _{q,\alpha,\beta } \leq m^{2\nu -1/q}. $$
Hence, for \(1 \leq q\leq \infty \) and \(F_{a}= \sum^{N}_{j=1}a_{j} \varphi_{j} \in A_{N}\),
$$\bigl\Vert (-(D_{\alpha,\beta })^{\nu }(F_{a})\bigr\Vert _{q,\alpha,\beta } \leq m ^{2\nu -1/q} \Vert a\Vert _{l^{N}_{q}}. $$
It then follows by the Kolmogorov type inequality (see Theorem 8.1 in [18]) that
$$\begin{aligned} \bigl\Vert F^{(\rho)}_{a}\bigr\Vert _{q,\alpha,\beta } =& \bigl\Vert (-D_{\alpha,\beta })^{\rho /2}(F_{a}) \bigr\Vert _{q,\alpha,\beta } \\ \ll& \bigl\Vert (-D_{\alpha,\beta })^{1+[\rho ]}(F_{a}) \bigr\Vert ^{\frac{\rho }{2+2[\rho ]}}_{q,\alpha,\beta } \Vert F_{a}\Vert ^{1-\frac{\rho }{2+2[\rho ]}}_{q,\alpha,\beta } \\ \ll& m^{\rho -1/q} \Vert a\Vert _{l^{N}_{q}} \ll m^{\rho } \Vert F_{a}\Vert _{q,\alpha,\beta }. \end{aligned}$$
(4.5)
For \(f\in L_{1,\alpha,\beta }\) and \(x \in [-1,1]\), we define
$$P_{N}(f) (x)=\sum^{N}_{j=1} \frac{\varphi_{j}(x)}{\Vert \varphi_{j}\Vert ^{2} _{2,\alpha,\beta }} \int^{1}_{-1} f(y)\varphi_{j}(y)W_{\alpha, \beta }(y)\,dy $$
and
$$Q_{N}(f) (x)=\sum^{N}_{j=1} \frac{\varphi_{j}(x)}{\Vert \varphi_{j}\Vert ^{2} _{2,\alpha,\beta }} \int^{1}_{-1} f(y)\varphi^{(\rho)}_{j}(y)W_{ \alpha,\beta }(y)\,dy. $$
Clearly, the operator \(P_{N}\) is the orthogonal projector from \(L_{2,\alpha,\beta }\) to \(A_{N}\), and if \(f\in W_{2,\alpha,\beta } ^{\rho }\), then \(Q_{N}(f)(x)=P_{N}(f^{\rho })(x)\). Also, using the method in [19], we can prove that \(P_{N}\) is the bounded operator from \(L_{q,\alpha,\beta }\) to \(A_{N}\cap L_{q,\mu }\) for \(1\leq q \leq \infty \),
$$ \bigl\Vert P_{N}(f)\bigr\Vert _{q,\alpha,\beta }\ll \Vert f \Vert _{q,\alpha,\beta }. $$
(4.6)
Since \(Q_{N}(f)\in A_{N}\) for \(f\in W_{2,\alpha,\beta }^{\rho }\), we have
$$ \bigl\Vert Q_{N}(f)^{(\rho)}\bigr\Vert _{2,\alpha,\beta }\ll m^{\rho }\bigl\Vert Q_{N}(f))\bigr\Vert _{2,\alpha,\beta } =m^{\rho }\bigl\Vert P_{N}(f)^{(\rho)}\bigr\Vert _{2,\alpha, \beta }\ll m^{\rho }\bigl\Vert f^{(\rho)}\bigr\Vert _{2,\alpha,\beta }. $$
(4.7)

Theorem 4.2

Let \(1\le q\le \infty \), \(\delta \in (0,1)\), and let N be given above. Then
$$\lambda_{n,\delta }\bigl(W^{r}_{2,\alpha,\beta },\nu,L_{q,\alpha,\beta }\bigr)\gg n^{1/2-\rho -1/q}\lambda_{n,\delta } \bigl(I_{N}:\mathbb{R}^{N}\rightarrow l^{N}_{q}, \gamma_{N}\bigr), $$
where \(N\asymp n\), \(N\geq 2n\) and \(\gamma_{N}\) is the standard Gaussian measure in \(\mathbb{R}^{N}\).

Proof

Let \(T_{n}\) be a bounded linear operator on \(W^{r}_{2,\alpha,\beta }\) with rank \(T_{n}\leq n\) such that
$$\nu \bigl(\bigl\{ f\in W^{r}_{2,\alpha,\beta }:\Vert f-T_{n} f\Vert _{q,\alpha,\beta }>2 \lambda_{n,\delta }\bigr\} \bigr)\leq \delta, $$
where \(\lambda_{n,\delta }:=\lambda_{n,\delta }(W^{r}_{2,\alpha, \beta },\nu,L_{q,\alpha,\beta })\). Note that if A is a bounded linear operator from \(W^{r}_{2,\alpha,\beta }\) to \(W^{r}_{2,\alpha, \beta }\) and from \(H(\nu)\) to \(H(\nu)\), then the image measure λ of ν under A is also a centered Gaussian measure on \(W^{r}_{2,\alpha,\beta }\) with covariance
$$R_{\lambda }(f) (f)=\bigl\langle A^{*}C_{\nu }f,A^{*}C_{\nu }f \bigr\rangle _{H(\nu)}, \quad f\in W^{r}_{2,\alpha,\beta }, $$
where \(C_{\nu }\) is the covariance of the measure ν, \(H(\nu)=W ^{\rho }_{2,\alpha,\beta }\) is the Camera-Martin space of ν, and \(A^{*}\) is the adjoint of A in \(H(\nu)\) (see Theorem 3.5.1 of [10]). Furthermore, if the operator A also satisfies
$$\Vert Af\Vert _{H(\nu)}\le \Vert f\Vert _{H(\nu)}, $$
then
$$R_{\lambda }(f) (f)=\bigl\Vert A^{*}C_{\nu }f\bigr\Vert ^{2}_{H(\nu)}\le \bigl\Vert A^{*}\bigr\Vert ^{2} \Vert C_{\nu }f\Vert \le \langle C_{\nu }f,C_{\nu }f\rangle_{H(\nu)}=R_{ \nu }(f) (f). $$
By Theorem 3.3.6 in [10], we get that for any absolutely convex Borel set E of \(W^{r}_{2,\alpha,\beta }\) there holds the inequality
$$\nu (E)\le \lambda (E). $$
Applying (4.7) we assert that
$$\bigl\Vert Q_{N}(f)\bigr\Vert _{H(\nu)}=\bigl\Vert \bigl(Q_{N}(f)\bigr)^{(\rho)}\bigr\Vert _{2,\alpha,\beta } \ll m^{\rho }\bigl\Vert f^{(\rho)}\bigr\Vert _{2,\alpha,\beta }=m^{\rho } \Vert f\Vert _{H(\nu)}. $$
Then there exists a positive constant \(c_{3}\) such that
$$\biggl\Vert \frac{1}{c_{3} m^{\rho }}Q_{N}(f)\biggr\Vert _{H(\nu)} \leq \Vert f\Vert _{H(\nu)}. $$
Note that, for any \(t>0\), the set \(\{f\in W^{r}_{2,\alpha,\beta }:\Vert f-T_{n} f\Vert _{q,\alpha,\beta }\leq t\}\) is absolutely convex. It then follows that
$$\nu \bigl(\bigl\{ f\in W^{r}_{2,\alpha,\beta }:\Vert f-T_{n}f\Vert _{q,\alpha,\beta }< 2 \lambda_{n,\delta }\bigr\} \bigr)\leq \lambda \bigl(\bigl\{ f\in W^{r}_{2,\alpha,\beta }: \Vert f-T_{n}f\Vert _{q,\alpha,\beta }< 2\lambda_{n,\delta }\bigr\} \bigr), $$
which leads to
$$\begin{aligned}& \nu \bigl(\bigl\{ f\in W^{r}_{2,\alpha,\beta }:\Vert f-T_{n}f\Vert _{q,\alpha,\beta }>2 \lambda_{n,\delta }\bigr\} \bigr) \\& \quad \geq \nu \bigl(\bigl\{ f\in W^{r}_{2,\alpha,\beta }:\Vert Q_{N} f - T_{n} Q_{N} f\Vert _{q,\alpha,\beta }>2 c_{3} m^{\rho } \lambda_{n,\delta } \bigr\} \bigr). \end{aligned}$$
Let \(L_{N} : \mathbb{R}^{N} \rightarrow A_{N}\) and \(J_{N} : A_{N} \rightarrow \mathbb{R}^{N}\) be defined by
$$L_{N}(a) (x)=\sum^{N}_{i=1} \frac{a_{i} \varphi_{i}(x)}{\Vert \varphi_{i}\Vert _{2,\alpha,\beta }}, \quad a=(a_{1},\ldots,a_{N})\in \mathbb{R}^{N} $$
and
$$J_{N}(F_{a})=\bigl(a_{1} \Vert \varphi_{1}\Vert _{2,\alpha,\beta },\ldots,a_{N} \Vert \varphi_{N}\Vert _{2,\alpha,\beta }\bigr),\quad F_{a} \in A_{N}. $$
We see at once that \(L_{N}J_{N}(F_{a})=F_{a}\) for any \(F_{a}\in A_{N}\). Set \(y=(y_{1},\ldots,y_{N})\in \mathbb{R}^{N} \), where \(y_{j} = \frac{1}{ \Vert \varphi_{j}\Vert _{2,\alpha,\beta }}\langle f, \varphi_{j}^{(\rho)} \rangle \). Then \(y=J_{N} Q_{N} (f)\). Thus by (4.4) and \(\Vert \varphi_{j}\Vert _{2,\alpha,\beta }\asymp m^{-\frac{1}{2}}\), we obtain
$$ \bigl\Vert L_{N}(a)\bigr\Vert _{q,\alpha,\beta }\asymp m^{-\frac{1}{q}+\frac{1}{2}}\Vert a\Vert _{ l_{q}^{N}}. $$
(4.8)
Combining (4.6) with (4.8), we conclude that for any \(f\in W^{r}_{2, \alpha,\beta }\),
$$\begin{aligned} \bigl\Vert Q_{N}(f)-T_{N}Q_{N} (f)\bigr\Vert _{q,\alpha,\beta } \gg &\bigl\Vert P_{N}\bigl(Q_{N}(f) \bigr)-P_{N} T_{n}Q_{N}(f)Q\bigr\Vert _{q,\alpha,\beta } \\ =&\bigl\Vert L_{N}J_{N}Q_{N}(f)-L_{N}J_{N}P_{N}T_{N}L_{N}J_{N}Q_{N}(f) \bigr\Vert _{q, \alpha,\beta } \\ \gg & m^{-\frac{1}{q}+\frac{1}{2}} \bigl\Vert J_{N}Q_{N}(f)-J_{N}P_{N}T_{n}L_{N}J_{N}Q_{N}(f) \bigr\Vert _{l^{N}_{q}} \\ \gg & m^{-\frac{1}{q}+\frac{1}{2}} \Vert y-J_{N}P_{N}T_{n}L_{N}y \Vert _{l ^{N}_{q}}. \end{aligned}$$
Remark that \(g_{k}=\frac{\varphi_{k}}{\Vert \varphi_{k}\Vert _{2,\alpha, \beta }}\), \(k=1,2,\ldots,N\), is an orthonormal system in \(L_{2,\alpha,\beta }\) and \(g_{k}\in H(v)=W^{\rho }_{2,\alpha,\beta }\). Then the random vector \((\langle f,g^{(\rho)}_{1}\rangle,\ldots,\langle f,g ^{(\rho)}_{N}\rangle)=y\) in \(\mathbb{R}^{N}\) on the measurable space \((W^{r}_{2,\alpha,\beta },\nu)\) has the standard Gaussian distribution \(r_{N}\) in \(\mathbb{R}^{N}\). It then follows that
$$\begin{aligned}& \nu \bigl(\bigl\{ f\in W^{r}_{2,\alpha,\beta }:\bigl\Vert Q_{N}(f)-T_{n}Q_{N}(f)\bigr\Vert _{q, \alpha,\beta }>2c_{3}m^{\rho }\lambda_{n,\delta }\bigr\} \bigr) \\& \quad \geq \nu \bigl(\bigl\{ f\in W^{r}_{2,\alpha,\beta }:\Vert y-T_{J}NP_{N}T_{n}L_{N}y\Vert _{l^{N}_{q}} >c_{4}m^{\rho +\frac{1}{q}-\frac{1}{2}} \lambda_{n, \delta }\bigr\} \bigr) \\& \quad =r_{N}\bigl(\bigl\{ y\in \mathbb{R}^{N}:\Vert y-T_{J}NP_{N}T_{n}L_{N}y\Vert _{l^{N}_{q}} >c _{4}m^{\rho +\frac{1}{q}-\frac{1}{2}} \lambda_{n,\delta }\bigr\} \bigr) \\& \quad =:r_{N}(G), \end{aligned}$$
where \(c_{4}\) is a positive constant. Clearly, \(\operatorname{rank}(J_{N}P_{N}T_{n}L _{N})\leq n\) and
$$r_{N}(G)\leq \nu \bigl(\bigl\{ f\in W^{r}_{2,\alpha,\beta }: \Vert f-T_{n}f\Vert _{q, \alpha,\beta }>2\lambda_{n,\delta }\bigr\} \bigr) \leq \delta. $$
Consequently,
$$\begin{aligned} \lambda_{n,\delta }\bigl(I_{N}:\mathbb{R}^{N} \rightarrow l^{N}_{q},r_{N}\bigr) =& \inf _{G} \inf_{I_{N}} \sup_{x\in \mathbb{R}^{N}\setminus G} \Vert I_{N}x-T_{n}x\Vert _{l^{N}_{q}} \\ \leq & \sup_{y\in \mathbb{R}^{N}\setminus G}\Vert I_{N}y-J_{N}P_{N}T_{n}L_{N}y \Vert _{l^{N}_{q}} \\ \ll & m^{\rho +\frac{1}{q}-\frac{1}{2}}\lambda_{n,\delta }, \end{aligned}$$
which implies
$$\begin{aligned} \lambda_{n,\delta }\bigl(W^{r}_{2,\alpha,\beta },\nu,L_{q,\alpha,\beta }\bigr) \ll & m^{-\rho -\frac{1}{q}+\frac{1}{2}}\lambda_{n,\delta } \bigl(I_{N}: \mathbb{R}^{N}\rightarrow l^{N}_{q},r_{N} \bigr) \\ \asymp & n^{-\rho -\frac{1}{q}+\frac{1}{2}}\lambda_{n,\delta }\bigl(I_{N}: \mathbb{R}^{N}\rightarrow l^{N}_{q},r_{N} \bigr). \end{aligned}$$
This completes the proof of Theorem 4.2. □

Now, we are in a position to prove Theorem 2.1.

Proof

For the lower estimates, using Theorem 4.2 and Lemma 3.1, we have for \(1\le q\le 2\)
$$\begin{aligned} \lambda_{n,\delta }\bigl(W_{2,\alpha,\beta }^{r},\nu,L_{q,\alpha,\beta }\bigr) \gg & n^{-\rho +1/2-1/q}\lambda_{n,\delta } \bigl(I_{N}:\mathbb{R}^{N} \rightarrow l_{q}^{N}, \gamma_{N}\bigr) \\ \asymp & n^{-\rho +1/2-1/q} \biggl(N^{1/q}+N^{1/q-1/2}\biggl(\ln \biggl(\frac{1}{ \delta }\biggr)\biggr)^{1/2} \biggr) \\ \asymp & n^{1/2-\rho } \biggl(1+n^{-1/2}\biggl(\ln \biggl(\frac{1}{\delta }\biggr)\biggr)^{1/2} \biggr). \end{aligned}$$
For \(2\le q< \infty \), we have
$$\begin{aligned} \lambda_{n,\delta }\bigl(W_{2,\alpha,\beta }^{r},\nu,L_{q,\alpha,\beta }\bigr) \gg & n^{-\rho +1/2-1/q} \biggl(n^{1/q}+\biggl(\ln \biggl(\frac{1}{\delta }\biggr)\biggr)^{1/2} \biggr) \\ \asymp & n^{1/2-\rho } \biggl(1+n^{-1/q}\biggl(\ln \biggl(\frac{1}{\delta }\biggr)\biggr)^{1/2} \biggr). \end{aligned}$$
And for \(q=\infty \),
$$\begin{aligned} \lambda_{n,\delta }\bigl(W_{2,\alpha,\beta }^{r},\nu,L_{q,\alpha,\beta }\bigr) \gg & n^{-\rho +1/2-1/q} \biggl(\ln m+\ln \biggl(\frac{1}{\delta }\biggr) \biggr)^{1/2} \\ =& n^{1/2-\rho } \biggl(\ln \biggl(\frac{m}{\delta }\biggr) \biggr)^{1/2} . \end{aligned}$$
It remains to prove the upper estimates. For \(2\le q\le \infty \) and any fixed natural number n, assume \(C_{1}2^{m}\le n\le C_{1}^{2}2^{m}\) with \(C_{1}>0\) to be specified later. We may take sufficiently small positive numbers \(\varepsilon >0\) such that \(\rho >\frac{1}{2}+(1+ \varepsilon)(2\max \{\alpha,\beta \}+1+\varepsilon)(\frac{1}{2}- \frac{1}{q})\). Set
$$n_{j}=\textstyle\begin{cases} 2^{j+1}, & \mbox{if } j\le m, \\ 2^{j+1}2^{(1+\varepsilon)(m-j)-1}, & \mbox{if } j> m, \end{cases} $$
and
$$\delta_{j}=\textstyle\begin{cases} 0, & \mbox{if } j\le m, \\ \delta 2^{m-j}, & \mbox{if } j> m. \end{cases} $$
Then
$$\sum_{j\ge 0}n_{j}\ll \sum _{j\le m}2^{j}+\sum_{j>m}2^{m(1+\varepsilon)-\varepsilon j} \ll 2^{m} $$
and
$$\sum_{j\ge 0}\delta_{j}=\delta \sum _{j\le m}2^{m-j}\le \delta. $$
Thus, we can take \(C_{1}\) sufficiently large so that
$$\sum_{j=0}^{\infty }n_{j}\le C_{1}2^{m}\le n. $$
It follows from Lemma 3.4 for \(\tau \in (0,\frac{1}{(2\max \{\alpha, \beta \}+1)(1/2-1/q)})\), \(2\le q\le \infty \),
$$\sum_{j=1}^{n} b_{j}^{-\tau (1/2-1/q)} \ll 2^{k[1+\tau (1/2-1/q)]}=2^{k+k \tau (1/2-1/q)}. $$
If \(j\le m\), then \(n_{j}=2^{j+1}\), and thence \(\lambda_{n_{j},\delta _{j}}(V_{j}:\mathbb{R}^{2^{j+1}}\rightarrow l_{q}^{2^{j+1}},\gamma_{2^{j+1}})=0\). If \(j>m\), then taking \(\frac{1}{\tau }=(2\max \{\alpha,\beta \}+1+ \varepsilon)(1/2-1/q)\) and applying Lemma 3.2, Theorem 4.1, we obtain for \(2\le q<\infty \),
$$\begin{aligned}& \lambda_{n_{j},\delta_{j}}\bigl(V_{j}:\mathbb{R}^{2^{j+1}} \rightarrow l_{q} ^{2^{j+1}},\gamma_{2^{j+1}}\bigr) \\& \quad \ll \biggl(\frac{C(m,\tau)}{n_{j}+1} \biggr)^{1/ \tau } \biggl(2^{(j+1)/q}+\sqrt{\ln \frac{1}{\delta }} \biggr) \\& \quad \ll 2^{j(1/2-1/q)}2^{-(1+\varepsilon)(m-j)(2\max \{\alpha,\beta \}+1+\varepsilon)(\frac{1}{2}-\frac{1}{q})} \biggl(2^{\frac{j}{q}}+\sqrt{ \ln \frac{1}{\delta }} \biggr), \end{aligned}$$
which yields
$$\begin{aligned}& \lambda_{n,\delta }\bigl(W_{2,\alpha,\beta }^{r},\nu,L_{q,\alpha,\beta }\bigr) \\& \quad \ll \sum_{j=m+1}^{\infty }2^{-j\rho }2^{j(1/2-1/q)}2^{-(1+\varepsilon)(m-j)(2\max \{\alpha,\beta \}+1+\varepsilon)(\frac{1}{2}- \frac{1}{q})}2^{1/2-1/q} \biggl(2^{\frac{j}{q}}+\sqrt{\ln \frac{1}{ \delta }} \biggr) \\& \quad \ll 2^{-m(\rho -\frac{1}{2}+\frac{1}{q})} \biggl(2^{\frac{m}{q}}+\sqrt{ \ln \frac{1}{\delta }} \biggr) \asymp n^{1/2-\rho } \biggl(1+n^{-1/q}\sqrt{ \ln \frac{1}{\delta }} \biggr). \end{aligned}$$
(4.9)
For \(q=\infty \), by Lemma 3.2 we get
$$\begin{aligned} \lambda_{n_{j},\delta_{j}}\bigl(V_{j}:\mathbb{R}^{2^{j+1}} \rightarrow l_{q} ^{2^{j+1}},\gamma_{2^{j+1}}\bigr) \ll & \biggl(\frac{C(2^{j+1},\tau)}{n_{j}+1} \biggr)^{1/\tau }\sqrt{\ln 2^{j+1}+ \ln \frac{1}{\delta }} \\ =&2^{j/2-(1+\varepsilon)(m-j)(2\max \{\alpha,\beta \}+1+\varepsilon)/2}\sqrt{j+\ln \frac{1}{\delta }}, \end{aligned}$$
then applying Theorem 4.1, we obtain
$$\begin{aligned} \lambda_{n,\delta }\bigl(W_{2,\alpha,\beta }^{r},\nu,L_{\infty,\alpha, \beta }\bigr) &\ll \sum_{j=m+1}^{\infty }2^{-j\rho }2^{j/2-(1+\varepsilon)(m-j)(2 \max \{\alpha,\beta \}+1+\varepsilon)/2} \sqrt{j+\ln \frac{1}{ \delta }} \\ &\ll 2^{-m(\rho -\frac{1}{2})}\sqrt{m+\ln \frac{1}{\delta }} \asymp n^{1/2-\rho } \sqrt{\ln \frac{n}{\delta }}. \end{aligned}$$
(4.10)
To finish the proof of the upper estimates, we only need to show that, for \(1\le q<2\),
$$\lambda_{n,\delta }\bigl(W_{2,\alpha,\beta }^{r},\nu,L_{q,\alpha,\beta }\bigr)\ll \lambda_{n,\delta }\bigl(W_{2,\alpha,\beta }^{r}, \nu,L_{2,\alpha, \beta }\bigr) \ll n^{1/2-\rho } \biggl(1+n^{-1}\sqrt{ \ln \frac{1}{\delta }} \biggr)^{1/2}. $$
Theorem 2.1 is proved. □

5 Conclusions

In this paper, optimal estimates of the probabilistic linear \((n,\delta)\)-widths of the weighted Sobolev space \(W_{2,{\alpha, \beta }}^{r}\) on \([-1,1]\) 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.

Declarations

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Project no. 11401520), by the Research Award Fund for Outstanding Young and Middle-aged Scientists of Shandong Province (BS2014SF019), by the National Natural Science Foundation of China (11271263), by the Beijing Natural Science Foundation (1132001), and BCMIIS.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All the authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematical and Statistics, Zaozhuang University

References

  1. Fang, G, Ye, P: Probabilistic and average linear widths of Sobolev space with Gaussian measure. J. Complex. 19, 73-84 (2003) View ArticleMATHMathSciNetGoogle Scholar
  2. Fang, G, Ye, P: Probabilistic and average linear widths of Sobolev space with Gaussian measure in L∞-norm. Constr. Approx. 20, 159-172 (2004) MATHMathSciNetGoogle Scholar
  3. Maiorov, VE: Widths of spaces equipped with a Gaussian measure. Russ. Acad. Sci. Dokl. Math. 45(2), 305-309 (1993) MathSciNetGoogle Scholar
  4. Maiorov, VE: Linear widths of function spaces equipped with the Gaussian measure. J. Approx. Theory 77(1), 74-88 (1994) View ArticleMATHMathSciNetGoogle Scholar
  5. Maiorov, VE, Wasilkowski, GW: Probabilistic and average linear widths in L∞-norm with respect to r-fold Wiener measure. J. Approx. Theory 84(1), 31-40 (1996) View ArticleMATHMathSciNetGoogle Scholar
  6. Wang, H, Zhai, X: Approximation of functions on the sphere on a Sobolev space with a Gaussian measure in the probabilistic case setting. Int. J. Wavelets Multiresolut. Inf. Process. 12(5), 1461012 (2014) View ArticleMATHMathSciNetGoogle Scholar
  7. Wang, H: Probabilistic and average widths of weighted Sobolev spaces on the ball equipped with a Gaussian measure (2016) https://arxiv.org/abs/1603.04578
  8. Szegö, G: Orthogonal Polynomials, 4th edn., vol. 23. Amer. Math. Soc., Providence (1975) MATHGoogle Scholar
  9. Dunkl, CF, Xu, Y: Orthogonal Polynomials of Several Variables. Cambridge University Press, Cambridge (2001) View ArticleMATHGoogle Scholar
  10. Bogachev, VI: Gaussian Measures. Mathematical Surveys and Monographs, vol. 62. Am. Math. Soc., Providence (1998) MATHGoogle Scholar
  11. Kuo, HH: Gaussian Measure in Banach Space. Lecture Notes in Mathematics, vol. 463. Springer, Berlin (1975) View ArticleGoogle Scholar
  12. Zhai, X, Hu, X: Approximation of functions on \([-1,1]\) with Jacobi weights in the worst and average case settings. Sci. Sin., Math. 44(2), 165-182 (2014) (in Chinese) View ArticleGoogle Scholar
  13. Maiorov, VE: Discretization of a problem about n-widths. Usp. Mat. Nauk 30, 179-180 (1976) Google Scholar
  14. Maiorov, VE: Kolmogorov’s \((n, \delta)\)-widths of spaces of the smooth functions. Russ. Acad. Sci. Sb. Math. 79(2), 265-279 (1994) MathSciNetGoogle Scholar
  15. Frenzen, CL, Wong, R: A uniform asymptotic expansion of the Jacobi polynomials with error bounds. Can. J. Math. 37, 979-1007 (1985) View ArticleMATHMathSciNetGoogle Scholar
  16. Nevai, P: Orthogonal Polynomials. Memoirs of AMS, vol. 18 (1979) MATHGoogle Scholar
  17. Szabados, J, Vértesi, P: Interpolation of Functions. World Scientific, Singapore (1990) View ArticleMATHGoogle Scholar
  18. Ditzan, Z: Fractional derivatives and best approximation. Acta Math. Hung. 81(4), 323-348 (1998) View ArticleMathSciNetGoogle Scholar
  19. Wang, H, Huang, H: Widths of weighted Sobolev classes on the ball. J. Approx. Theory 154(2), 126-139 (2008) View ArticleMATHMathSciNetGoogle Scholar

Copyright

© The Author(s) 2017