# 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

$\mathrm{T}=\left\{\mathrm{x}=\left({x}_{1},{x}_{2},\dots ,{x}_{d}\right):{x}_{i}\ge 0,i=1,2,\dots ,d,|\mathrm{x}|=\sum _{i=1}^{d}{x}_{i}\le 1\right\}.$

Let ${L}_{p}\left(\mathrm{T}\right):={L}_{\mathrm{p}}\left(\mathrm{T}\right)$, $\mathrm{p}=\left({p}_{1},{p}_{2},\dots ,{p}_{d}\right)$, ${p}_{1}={p}_{2}=\cdots ={p}_{d}=p$, $1\le p<\mathrm{\infty }$, be the space consisting of all Lebesgue measurable functions f on T for which the norm ${\parallel f\parallel }_{p}:={\left({\int }_{\mathrm{T}}{|f\left(\mathrm{x}\right)|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}$ is finite. Let $C\left(\mathrm{T}\right):={L}_{\mathrm{\infty }}\left(\mathrm{T}\right)$, $\mathrm{\infty }=\left(\mathrm{\infty },\mathrm{\infty },\dots ,\mathrm{\infty }\right)$ be the space consisting of all continuous functions f on T for which the norm ${max}_{\mathrm{x}\in \mathrm{T}}|f\left(\mathrm{x}\right)|$ is finite.

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

${M}_{n,\mathrm{d}}\left(f;\mathrm{x}\right)=\sum _{|\mathrm{k}|\le n}{p}_{n,\mathrm{k}}\left(\mathrm{x}\right)\frac{\left(n+d\right)!}{n!}{\int }_{\mathrm{T}}{p}_{n,\mathrm{k}}\left(\mathrm{u}\right)f\left(\mathrm{u}\right)\phantom{\rule{0.2em}{0ex}}d\mathrm{u},$
(1.1)

where

${p}_{n,\mathrm{k}}\left(\mathrm{x}\right)=\frac{n!}{\mathrm{k}!\left(n-|\mathrm{k}|\right)!}{\mathrm{x}}^{\mathrm{k}}{\left(1-\mathrm{x}\right)}^{n-|\mathrm{k}|},\phantom{\rule{1em}{0ex}}\mathrm{x}\in \mathrm{T},$

$\mathrm{x}=\left({x}_{1},{x}_{2},\dots ,{x}_{d}\right)\in {\mathbb{R}}^{d}$, $\mathrm{k}=\left({k}_{1},{k}_{2},\dots ,{k}_{d}\right)\in {\mathbb{N}}_{0}^{d}$, ${\mathbb{N}}_{0}^{d}=\stackrel{d}{\stackrel{⏟}{{\mathbb{N}}_{0}×{\mathbb{N}}_{0}×\cdots ×{\mathbb{N}}_{0}}}$, ${\mathbb{N}}_{0}=\mathbb{N}\cup \left\{0\right\}$, $|x|={\sum }_{i=1}^{d}{x}_{i}$, ${\mathrm{x}}^{\mathrm{k}}={x}_{1}^{{k}_{1}}{x}_{2}^{{k}_{2}}\cdots {\mathrm{x}}_{d}^{{k}_{d}}$, $|\mathrm{k}|={\sum }_{i=1}^{d}{k}_{i}$, $\mathrm{k}!={k}_{1}!{k}_{2}!\cdots {k}_{d}!$.

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

Let ${D}_{i}={D}_{ii}=\frac{\partial }{\partial {x}_{i}}$, $1\le i\le d$; ${D}_{ij}={D}_{i}-{D}_{j}$, $1\le i; ${D}^{k}={D}_{1}^{{k}_{1}}{D}_{2}^{{k}_{2}}\cdots {D}_{d}^{{k}_{d}}$, $\mathrm{k}\in {N}_{0}^{d}$, and

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

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

${K}_{\phi }^{2}{\left(f;{t}_{l}^{2}\right)}_{p}=\underset{g\in {W}_{\mathrm{\Phi },p}^{2}}{inf}\left\{{\parallel f-g\parallel }_{p}+{t}_{l}^{2}\mathrm{\Phi }{\left(g\right)}_{p}\right\},\phantom{\rule{1em}{0ex}}{t}_{l}>0,l=1,2,\dots ,d,$

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

The anisotropic Besov spaces  are given by

${B}_{p,q}^{\frac{\theta }{2}}:={\left({L}_{p},{W}_{\mathrm{\Phi },p}^{2}\right)}_{\frac{\theta }{2},q},$

where $\theta =\left({\theta }_{1},{\theta }_{2},\dots ,{\theta }_{d}\right)$, $1\le p,q<\mathrm{\infty }$, $n\in \mathbb{N}$, $n>2>{\theta }_{l}>0$.

By  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

$f\in {B}_{p,q}^{\frac{\theta }{2}}\phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}{\int }_{0}^{\mathrm{\infty }}{\left[{t}_{l}^{-\frac{{\theta }_{l}}{2}}{K}_{\phi }^{2}{\left(f;{t}_{l}^{2}\right)}_{p}\right]}^{q}\frac{d{t}_{l}}{{t}_{l}}<\mathrm{\infty },$
(1.2)

and

${\int }_{0}^{\mathrm{\infty }}{\left[{t}_{l}^{-\frac{{\theta }_{l}}{2}}{K}_{\phi }^{2}{\left(f;{t}_{l}^{2}\right)}_{p}\right]}^{q}\frac{d{t}_{l}}{{t}_{l}}<\mathrm{\infty }\phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}{\int }_{0}^{1}{\left[{t}_{l}^{-\frac{{\theta }_{l}}{2}}{K}_{\phi }^{2}{\left(f;{t}_{l}^{2}\right)}_{p}\right]}^{q}\frac{d{t}_{l}}{{t}_{l}}<\mathrm{\infty }.$
(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

$f\in {B}_{p,q}^{\frac{\theta }{2}}\phantom{\rule{1em}{0ex}}⇔\phantom{\rule{1em}{0ex}}{\left\{\sum _{n=1}^{\mathrm{\infty }}{\left[{n}^{\frac{{\theta }_{l}}{2}}{\parallel {L}_{n}\left(f\right)-f\parallel }_{p}\right]}^{q}\frac{1}{n}\right\}}^{\frac{1}{q}}<\mathrm{\infty }.$

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 .

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

${\parallel {M}_{n,d}\left(f\right)\parallel }_{p}\le M{\parallel f\parallel }_{p},$
(2.1)
${\parallel {\phi }_{ij}^{2}{D}_{ij}^{2}{M}_{n,d}\left(f\right)\parallel }_{p}\le Mn{\parallel f\parallel }_{p},\phantom{\rule{1em}{0ex}}1\le i\le j\le d.$
(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

${\parallel {\phi }_{ij}^{2}{D}_{ij}^{2}{M}_{n,d}\left(f\right)\parallel }_{p}\le M{\parallel {\phi }_{ij}^{2}{D}_{ij}^{2}f\parallel }_{p},\phantom{\rule{1em}{0ex}}i=1,2,\dots ,d.$
(2.3)

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

${\parallel {M}_{n,d}\left(f\right)-f\parallel }_{p}\le M{K}_{\phi }^{2}{\left(f;{n}^{-1}\right)}_{p}.$
(2.4)

Proof Let $f\in {L}_{p}$, It is shown in  that there exists a constant $M>0$ such that

${M}^{-1}{\omega }_{\phi }^{2}{\left(f;{t}_{l}\right)}_{p}\le {K}_{\phi }^{\ast ,2}{\left(f;{t}_{l}^{2}\right)}_{p}\le M{\omega }_{\phi }^{2}{\left(f;{t}_{l}\right)}_{p},$

where ${\omega }_{\phi }^{2}{\left(f;{t}_{l}\right)}_{p}$ is the modulus of smoothness of Ditzian-Totik type defined by

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

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

${K}_{\phi }^{\ast ,2}{\left(f;{t}_{l}^{2}\right)}_{p}=\underset{g\in {W}_{\mathrm{\Phi },p}^{2}}{inf}\left\{{\parallel f-g\parallel }_{p}+{t}_{l}^{2}\sum _{1\le i\le j\le d}{\parallel {\phi }_{ij}^{2}{D}_{ij}^{2}g\parallel }_{p}\right\},\phantom{\rule{1em}{0ex}}{t}_{l}>0,l=1,2,\dots ,d.$

We notice that  for $f\in {L}_{p}$, we have

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

Thus, for $g\in {W}_{\mathrm{\Phi },p}^{2}$, by the definition of K-functional ${K}_{\phi }^{\ast ,2}{\left(f;{t}_{l}^{2}\right)}_{p}$, we have

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

According to the definition of K-functional ${K}_{\phi }^{2}{\left(f;{t}_{l}^{2}\right)}_{p}$, 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

$\mathrm{\Phi }{\left({M}_{n,d}\left(f\right)\right)}_{p}\le Mn{K}_{\phi }^{2}{\left(f;{n}^{-1}\right)}_{p}.$
(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

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

According to the definition of K-functional ${K}_{\phi }^{2}{\left(f;{t}_{l}^{2}\right)}_{p}$, 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

$\begin{array}{rcl}f\in {B}_{p,q}^{\frac{\theta }{2}}\phantom{\rule{1em}{0ex}}& ⇔& \phantom{\rule{1em}{0ex}}{\left\{\sum _{n=1}^{\mathrm{\infty }}{\left({n}^{\frac{{\theta }_{l}}{2}}{\parallel {M}_{n,d}\left(f\right)-f\parallel }_{p}\right)}^{q}\frac{1}{n}\right\}}^{\frac{1}{q}}<\mathrm{\infty }\\ ⇔& \phantom{\rule{1em}{0ex}}{n}^{-\frac{1}{q}}{n}^{\frac{{\theta }_{l}}{2}}\left({M}_{n,d}\left(f;\mathrm{x}\right)-f\left(\mathrm{x}\right)\right)\in {l}^{q}\left({L}_{p}\right).\end{array}$
(3.1)

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

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

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

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

$\left(1\right)\phantom{\rule{0.25em}{0ex}}{A}^{r-1}\le {n}_{r}<{A}^{r};\phantom{\rule{2em}{0ex}}\left(2\right)\phantom{\rule{0.25em}{0ex}}{\parallel {M}_{{n}_{r},d}\left(f\right)-f\parallel }_{p}=\underset{{A}^{r-1}\le m<{A}^{r}}{min}{\parallel {M}_{m,d}\left(f\right)-f\parallel }_{p}.$

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

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

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, we have (3.3) (3.4)

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

${\int }_{0}^{1}{\left[{t}_{l}^{-\frac{{\theta }_{l}}{2}}{K}_{\phi }^{2}{\left(f;{t}_{l}\right)}_{p}\right]}^{q}\frac{d{t}_{l}}{{t}_{l}}<\mathrm{\infty }.$

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.

## References

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2. Nikolskii SM: Approximation of Functions of Several Variables and Imbedding Theorem. Springer, Berlin; 1975.

3. Bergh J, Lörfstrom J: Interpolation Spaces. Springer, Berlin; 1976.

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## Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments, remarks and suggestions which greatly helped us to improve the presentation of this paper and make it more readable. Project supported by the Natural Science Foundation of China (Grant No. 10671019), the Zhejiang Provincial Natural Science Foundation (Grant No. LY12A01008), and the Cultivation fund of Taizhou University.

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Feng, G., Feng, Y. Relations between anisotropic Besov spaces and multivariate Bernstein-Durrmeyer operators. J Inequal Appl 2013, 202 (2013). https://doi.org/10.1186/1029-242X-2013-202 