# Probabilistic linear widths of Sobolev space with Jacobi weights on $$[-1,1]$$

## 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 $$(1-x)^{\alpha }(1+x)^{\beta }$$, $$\alpha,\beta > -1/2$$.

## 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.,

$$H(\nu)= \overline{W}_{2,{\alpha,\beta }}^{r+s/2}.$$

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

$$\nu \bigl(\bigl\{ f\in W^{r}_{2,\alpha,\beta }:\Vert f-T_{n}f\Vert _{q,\alpha,\beta }< 2 \lambda_{n,\delta }\bigr\} \bigr)\leq \lambda \bigl(\bigl\{ f\in W^{r}_{2,\alpha,\beta }: \Vert f-T_{n}f\Vert _{q,\alpha,\beta }< 2\lambda_{n,\delta }\bigr\} \bigr),$$

\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.

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## 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.

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Zhai, X., Hu, X. Probabilistic linear widths of Sobolev space with Jacobi weights on $$[-1,1]$$ . J Inequal Appl 2017, 262 (2017). https://doi.org/10.1186/s13660-017-1540-7