Relations between anisotropic Besov spaces and multivariate Bernstein-Durrmeyer operators

Abstract

In this paper, we use the multivariate Bernstein-Durrmeyer operators defined on the simplex to characterize anisotropic Besov spaces.

MSC:41A27, 41A36.

1 Introduction and some notations

Let T be the simplex in ${\mathbb{R}}^d$ defined by

$$\mathrm{T}=\{\mathrm{x}=(x_1,x_2,\dots ,x_d):x_i\ge 0,i=1,2,\dots ,d,|\mathrm{x}|=\sum _{i=1}^d{x}_i\le 1\}.$$

Let $L_p(\mathrm{T}):=L_{\mathrm{p}}(\mathrm{T})$, $\mathrm{p}=(p_1,p_2,\dots ,p_d)$, $p_1=p_2=\cdots =p_d=p$, $1\le p<\mathrm{\infty }$, be the space consisting of all Lebesgue measurable functions f on T for which the norm ${\parallel f\parallel }_p:={({\int }_{\mathrm{T}}{|f(\mathrm{x})|}^{p}dx)}^{1/p}$ is finite. Let $C(\mathrm{T}):=L_{\mathrm{\infty }}(\mathrm{T})$, $\mathrm{\infty }=(\mathrm{\infty },\mathrm{\infty },\dots ,\mathrm{\infty })$ be the space consisting of all continuous functions f on T for which the norm ${max}_{\mathrm{x}\in \mathrm{T}}|f(\mathrm{x})|$ is finite.

Let $f\in L_1(\mathrm{T})$. For each $n\in \mathbb{N}$, the multivariate Bernstein-Durrmeyer operators of f are defined by [1]

$$M_{n,\mathrm{d}}(f;\mathrm{x})=\sum _{|\mathrm{k}|\le n}{p}_{n,\mathrm{k}}(\mathrm{x})\frac{(n+d)!}{n!}{\int }_{\mathrm{T}}{p}_{n,\mathrm{k}}(\mathrm{u})f(\mathrm{u})d\mathrm{u},$$

(1.1)

where

$$p_{n,\mathrm{k}}(\mathrm{x})=\frac{n!}{\mathrm{k}!(n-|\mathrm{k}|)!}{\mathrm{x}}^{\mathrm{k}}{(1-\mathrm{x})}^{n-|\mathrm{k}|},\mathrm{x}\in \mathrm{T},$$

$\mathrm{x}=(x_1,x_2,\dots ,x_d)\in {\mathbb{R}}^d$, $\mathrm{k}=(k_1,k_2,\dots ,k_d)\in {\mathbb{N}}_0^d$, ${\mathbb{N}}_0^d=\stackrel{d}{\stackrel{⏟}{{\mathbb{N}}_0\times {\mathbb{N}}_0\times \cdots \times {\mathbb{N}}_0}}$, ${\mathbb{N}}_0=\mathbb{N}\cup \{0\}$, $|x|={\sum }_{i=1}^d{x}_i$, ${\mathrm{x}}^{\mathrm{k}}=x_1^{k_1}{x}_2^{k_2}\cdots {\mathrm{x}}_d^{k_d}$, $|\mathrm{k}|={\sum }_{i=1}^d{k}_i$, $\mathrm{k}!=k_1!k_2!\cdots k_d!$.

For $\mathrm{x}\in \mathrm{T}$, we denote

$${\phi }_{ij}^2(\mathrm{x})=\{\begin{array}{cc}{x}_i(1-|\mathrm{x}|)\hfill & \text{for }i=j=1,2,\dots ,d,\hfill \\ x_i{x}_j\hfill & \text{for }1\le i<j\le d.\hfill \end{array}$$

Let $D_i=D_{ii}=\frac{\partial }{\partial x_i}$, $1\le i\le d$; $D_{ij}=D_i-D_j$, $1\le i<j\le d$; $D^k=D_1^{k_1}{D}_2^{k_2}\cdots D_d^{k_d}$, $\mathrm{k}\in N_0^d$, and

$$D_{ij}^2(\mathrm{x})=\{\begin{array}{cc}\frac{{\partial }^2}{\partial x_i^2}\hfill & \text{for }i=j=1,2,\dots ,d,\hfill \\ {(\frac{\partial }{\partial x_i}-\frac{\partial }{\partial x_j})}^2\hfill & \text{for }1\le i<j\le d.\hfill \end{array}$$

Definition 1.1 Let $L_p:=L_p(\mathrm{T})$, $1\le p<\mathrm{\infty }$, and weighted Sobolev spaces are given by

$$W_{\mathrm{\Phi },p}^2=\{g\mid g\in L_p,D^{k}g,|k|\le 2\text{ are in }{L}_{\mathrm{loc}}(\stackrel{\mathrm{o}}{\mathrm{T}}),\text{ and }{\phi }_{ij}^2{D}_{ij}^{2}f\in L_p,1\le i\le j\le d\},$$

where the derivatives are in the sense of distributions, and $\stackrel{\mathrm{o}}{\mathrm{T}}$ is the interior of T.

The K-functional of Ditzian-Totik type is given by

$$K_{\phi }^2{(f;t_l^2)}_p=\underset{g\in W_{\mathrm{\Phi },p}^2}{inf}\{{\parallel f-g\parallel }_p+t_l^2\mathrm{\Phi }{(g)}_p\},t_l>0,l=1,2,\dots ,d,$$

where $\mathrm{t}=(t_1,t_2,\dots ,t_d)$, $\mathrm{\Phi }{(g)}_p:={\parallel g\parallel }_p+{\sum }_{1\le i\le j\le d}{\parallel {\phi }_{ij}^2{D}_{ij}^{2}g\parallel }_p$.

The anisotropic Besov spaces [2] are given by

$$B_{p,q}^{\frac{\theta }{2}}:={(L_p,W_{\mathrm{\Phi },p}^2)}_{\frac{\theta }{2},q},$$

where $\theta =({\theta }_1,{\theta }_2,\dots ,{\theta }_d)$, $1\le p,q<\mathrm{\infty }$, $n\in \mathbb{N}$, $n>2>{\theta }_l>0$.

By [3] and the definition of anisotropic Besov spaces, it is not difficult to get the following.

Theorem 1.2 Suppose $1\le p,q<\mathrm{\infty }$, $n\in \mathbb{N}$, $n>2>{\theta }_l>0$, $l=1,2,\dots ,d$. Then

$$f\in B_{p,q}^{\frac{\theta }{2}}\iff {\int }_0^{\mathrm{\infty }}{\left[t_l^{-\frac{{\theta }_l}{2}}{K}_{\phi }^2{(f;t_l^2)}_p\right]}^q\frac{d{t}_l}{t_l}<\mathrm{\infty },$$

(1.2)

and

$${\int }_0^{\mathrm{\infty }}{\left[t_l^{-\frac{{\theta }_l}{2}}{K}_{\phi }^2{(f;t_l^2)}_p\right]}^q\frac{d{t}_l}{t_l}<\mathrm{\infty }\iff {\int }_0^1{\left[t_l^{-\frac{{\theta }_l}{2}}{K}_{\phi }^2{(f;t_l^2)}_p\right]}^q\frac{d{t}_l}{t_l}<\mathrm{\infty }.$$

(1.3)

In this paper, we use the multivariate Bernstein-Durrmeyer operators defined on the simplex to characterize anisotropic Besov spaces. We will show, for $1\le p,q<\mathrm{\infty }$, $n\in \mathbb{N}$, $n>2>{\theta }_l>0$, that

$$f\in B_{p,q}^{\frac{\theta }{2}}\iff {\left\{\sum _{n=1}^{\mathrm{\infty }}{\left[n^{\frac{{\theta }_l}{2}}{\parallel L_n(f)-f\parallel }_p\right]}^q\frac{1}{n}\right\}}^{\frac{1}{q}}<\mathrm{\infty }.$$

For convenience, throughout this paper, M denotes a positive constant independent of x, n and f which may be different in different places.

2 Auxiliary lemmas

To prove the theorems, we need the following lemmas. The following two lemmas were proved in [4].

Lemma 2.1 If $1\le p<\mathrm{\infty }$, $f\in L_p$, $n\in \mathbb{N}$, then

$${\parallel M_{n,d}(f)\parallel }_p\le M{\parallel f\parallel }_p,$$

(2.1)

$${\parallel {\phi }_{ij}^2{D}_{ij}^2{M}_{n,d}(f)\parallel }_p\le Mn{\parallel f\parallel }_p,1\le i\le j\le d.$$

(2.2)

Lemma 2.2 If $1\le p<\mathrm{\infty }$, $f\in W_{\mathrm{\Phi },p}^2$, $n\in \mathbb{N}$, $n>2$, then

$${\parallel {\phi }_{ij}^2{D}_{ij}^2{M}_{n,d}(f)\parallel }_p\le M{\parallel {\phi }_{ij}^2{D}_{ij}^{2}f\parallel }_p,i=1,2,\dots ,d.$$

(2.3)

Lemma 2.3 Suppose $1\le p<\mathrm{\infty }$, $f\in L_p$, $n\in \mathbb{N}$, $n>2$. Then

$${\parallel M_{n,d}(f)-f\parallel }_p\le M{K}_{\phi }^2{(f;n^{-1})}_p.$$

(2.4)

Proof Let $f\in L_p$, It is shown in [5] that there exists a constant $M>0$ such that

$$M^{-1}{\omega }_{\phi }^2{(f;t_l)}_p\le K_{\phi }^{\ast ,2}{(f;t_l^2)}_p\le M{\omega }_{\phi }^2{(f;t_l)}_p,$$

where ${\omega }_{\phi }^2{(f;t_l)}_p$ is the modulus of smoothness of Ditzian-Totik type defined by

$$\begin{array}{r}{\omega }_{\phi }^2{(f;t_l)}_p:=\underset{0\le h\le t_l}{sup}\sum _{1\le i\le j\le d}{\parallel {\mathrm{\Delta }}_{h{\phi }_{ij}{e}_{ij}}^{2}f\parallel }_p,t_l>0,l=1,2,\dots ,d,\\ {\parallel {\mathrm{\Delta }}_{h\mathrm{e}}^{2}f(\mathrm{x})\parallel }_p=\{\begin{array}{cc}f(\mathrm{x}+\frac{he}{2})-2f(\mathrm{x}+\frac{he}{2})+f(\mathrm{x}-\frac{he}{2}),\hfill & \mathrm{x}\pm \frac{h\mathrm{e}}{2}\in \mathrm{T},\hfill \\ 0,\hfill & \text{otherwise},\hfill \end{array}\end{array}$$

$h>0$, $\mathrm{ei}=(0,0,\dots ,\stackrel{\mathrm{ith}}{1},0,\dots ,0)$ is the unit vector in ${\mathbb{R}}^d$, ${\mathrm{e}}_{\mathrm{i}\mathrm{j}}=\mathrm{ei}-\mathrm{ej}$, $\mathrm{e}\in {\mathbb{R}}^n$. $K_{\phi }^{\ast ,2}{(f;t_l^2)}_p$ is another K-functional of Ditzian-Totik type defined by

$$K_{\phi }^{\ast ,2}{(f;t_l^2)}_p=\underset{g\in W_{\mathrm{\Phi },p}^2}{inf}\{{\parallel f-g\parallel }_p+t_l^2\sum _{1\le i\le j\le d}{\parallel {\phi }_{ij}^2{D}_{ij}^{2}g\parallel }_p\},t_l>0,l=1,2,\dots ,d.$$

We notice that [6] for $f\in L_p$, we have

$${\parallel M_{n,d}(f)-f\parallel }_p\le M({\omega }_{\phi }^2{(f;\sqrt{n})}_p+n^{-1}{\parallel f\parallel }_p).$$

Thus, for $g\in W_{\mathrm{\Phi },p}^2$, by the definition of K-functional $K_{\phi }^{\ast ,2}{(f;t_l^2)}_p$, we have

$$\begin{array}{rcl}{\parallel M_{n,d}(f)-f\parallel }_p& \le & M({\omega }_{\phi }^2{(f;\sqrt{n})}_p+n^{-1}{\parallel f\parallel }_p)\\ \le & M(K_{\phi }^{\ast ,2}(f;n^{-\frac{1}{2}})+n^{-1}{\parallel f-g\parallel }_p+n^{-1}{\parallel g\parallel }_p)\\ \le & M(2{\parallel f-g\parallel }_p+n^{-1}{\parallel g\parallel }_p+n^{-1}\sum _{1\le i\le j\le d}{\parallel {\phi }_{ij}^2{D}_{ij}^{2}g\parallel }_p).\end{array}$$

According to the definition of K-functional $K_{\phi }^2{(f;t_l^2)}_p$, Lemma 2.3 has been proved. □

Lemma 2.4 Suppose $1\le p<\mathrm{\infty }$, $f\in L_p$, $n\in \mathbb{N}$, $n>2$. Then

$$\mathrm{\Phi }{(M_{n,d}(f))}_p\le Mn{K}_{\phi }^2{(f;n^{-1})}_p.$$

(2.5)

Proof For $f\in L_p$, $g\in W_{\mathrm{\Phi },p}^2$, by Lemma 2.1 and Lemma 2.2, we get

$$\begin{array}{rcl}\mathrm{\Phi }{(M_{n,d}(f))}_p& =& {\parallel M_{n,d}(f)\parallel }_p+\sum _{1\le i\le j\le d}{\parallel {\phi }_{ij}^2{D}_{ij}^2{M}_{n,d}(f)\parallel }_p\\ \le & {\parallel M_{n,d}(f-g)\parallel }_p+{\parallel M_{n,d}(g)\parallel }_p\\ +\sum _{1\le i\le j\le d}{\parallel {\phi }_{ij}^2{D}_{ij}^2{M}_{n,d}(f-g)\parallel }_p+\sum _{1\le i\le j\le d}{\parallel {\phi }_{ij}^2{D}_{ij}^2{M}_{n,d}(g)\parallel }_p\\ \le & M(n{\parallel f-g\parallel }_p+{\parallel g\parallel }_p+\sum _{1\le i\le j\le d}{\parallel {\phi }_{ij}^2{D}_{ij}^{2}g\parallel }_p)\\ \le & Mn({\parallel f-g\parallel }_p+n^{-1}({\parallel g\parallel }_p+\sum _{1\le i\le j\le d}{\parallel {\phi }_{ij}^2{D}_{ij}^{2}g\parallel }_p)).\end{array}$$

According to the definition of K-functional $K_{\phi }^2{(f;t_l^2)}_p$, Lemma 2.4 has been proved. □

3 Main results

In this section we will prove our main results.

Theorem 3.1 Let $1\le p,q<\mathrm{\infty }$, $n\in \mathbb{N}$, $n>2>{\theta }_l>0$, $l=1,2,\dots ,d$. Then

$$\begin{array}{rcl}f\in B_{p,q}^{\frac{\theta }{2}}& \iff & {\left\{\sum _{n=1}^{\mathrm{\infty }}{\left(n^{\frac{{\theta }_l}{2}}{\parallel M_{n,d}(f)-f\parallel }_p\right)}^q\frac{1}{n}\right\}}^{\frac{1}{q}}<\mathrm{\infty }\\ \iff & n^{-\frac{1}{q}}{n}^{\frac{{\theta }_l}{2}}(M_{n,d}(f;\mathrm{x})-f(\mathrm{x}))\in l^q(L_p).\end{array}$$

(3.1)

Proof First we prove the direct result of (3.1). By applying Lemma 2.3, we have

$$\begin{array}{rcl}\sum _{n=1}^{\mathrm{\infty }}{\left[n^{\frac{{\theta }_l}{2}}{\parallel M_{n,d}(f)-f\parallel }_p\right]}^q\frac{1}{n}& \le & \sum _{r=0}^{\mathrm{\infty }}\sum _{n=2^r}^{2^{r+1}-1}{\left[n^{\frac{{\theta }_l}{2}}M{K}_{\phi }^2{(f;n^{-1})}_p\right]}^q{n}^{-1}\\ \le & M\sum _{r=0}^{\mathrm{\infty }}{\left[n^{(r+1)\frac{{\theta }_l}{2}}{K}_{\phi }^2{(f;2^{-r})}_p\right]}^q\\ \le & M\frac{1}{ln2}{\left(2^{1+\frac{{\theta }_l}{2}}\right)}^q\sum _{r=0}^{\mathrm{\infty }}{\int }_{2^{-(r+1)}}^{2^{-r}}{\left[t^{-\frac{{\theta }_l}{2}}{K}_{\phi }^2{(f;t)}_p\right]}^q\frac{dt}{t}\\ \le & M\frac{1}{ln2}{\left(2^{1+\frac{{\theta }_l}{2}}\right)}^q{\int }_0^1{\left[t^{-\frac{{\theta }_l}{2}}{K}_{\phi }^2{(f;t)}_p\right]}^q\frac{dt}{t}.\end{array}$$

In virtue of $f\in B_{p,q}^{\frac{\theta }{2}}$ and by Theorem 1.2, we have

$$\sum _{n=1}^{\mathrm{\infty }}{\left[n^{\frac{{\theta }_l}{2}}{\parallel M_{n,d}(f)-f\parallel }_p\right]}^q\frac{1}{n}<\mathrm{\infty }.$$

(3.2)

The necessity has been proved.

Next, we prove the inverse result of (3.1). We take a constant $A\in \mathbb{N}$, which will be determined later. For $r\in \mathbb{N}$, we take $n_r\in \mathbb{N}$, which satisfies the following conditions:

$$(1)A^{r-1}\le n_r<A^r;(2){\parallel M_{n_r,d}(f)-f\parallel }_p=\underset{A^{r-1}\le m<A^r}{min}{\parallel M_{m,d}(f)-f\parallel }_p.$$

By using the definition of K-functional and Lemma 2.4, we derive by induction

$$\begin{array}{rcl}{A}^{\frac{\theta }{2}}{K}_{\phi }^2{(f;A^{-r})}_p& \le & A^{\frac{{\theta }_l}{2}}{\parallel f-M_{n_r,d}(f)\parallel }_p+M{A}^{(\frac{{\theta }_l}{2}-r)}{n}_r{K}_{\phi }^2{(f;n_r^{-1})}_p\\ \le & A^{\frac{{\theta }_l}{2}}{\parallel f-M_{n_r,d}(f)\parallel }_p+A^{r(\frac{{\theta }_l}{2}-1)}[M{n}_r{\parallel f-M_{n_{r-1},d}(f)\parallel }_p\\ +M^2{n}_{r-1}{K}_{\phi }^2{(f;n_{r-1}^{-1})}_p]\\ \le & \cdots \\ \le & A^{\frac{{\theta }_l}{2}}{\parallel f-M_{n_r,d}(f)\parallel }_p+A^{r(\frac{{\theta }_l}{2}-1)}[\sum _{v=1}^{r-1}{M}^l{n}_{r-v+1}{\parallel f-M_{n_{r-v},d}(f)\parallel }_p\\ +M^r{n}_1{K}_{\phi }^2{(f;n_1^{-1})}_p]\\ \le & A^{1+\frac{{\theta }_l}{2}}\sum _{v=1}^{r-1}{\left(M{A}^{v(\frac{{\theta }_l}{2}-1)}\right)}^v\left[n_{r-v}^{\frac{{\theta }_l}{2}}{\parallel f-M_{n_{r-v},d}(f)\parallel }_p\right]\\ +A{\left(M{A}^{\frac{{\theta }_l}{2}-1}\right)}^r{\parallel f\parallel }_p.\end{array}$$

We now choose $A\in \mathbb{N}$, $A\ge 2$, such that $\alpha :=M{A}^{\frac{{\theta }_l}{2}-1}<\frac{1}{2}$. For $1<q<\mathrm{\infty }$, we have

(3.3)

(3.4)

The proof for $q=1$ is easy and we shall omit it. Thus, we have

$${\int }_0^1{\left[t_l^{-\frac{{\theta }_l}{2}}{K}_{\phi }^2{(f;t_l)}_p\right]}^q\frac{d{t}_l}{t_l}<\mathrm{\infty }.$$

By Theorem 1.2, the sufficiency has also been proved. The proof is completed. □

Remark 1 For other integral-type operators, the method and the results are similar.