﻿ 二元(p, q)-Bernstein算子的逼近性质

# 二元(p, q)-Bernstein算子的逼近性质Approximation Properties of Bivariate (p, q)-Bernstein Operators

Abstract: In this paper, we introduce the bivariate (p, q)-Bernstein operator on the basis of (p, q)-Bernstein operator, and obtain the approximation theorem of the operator. The uniform convergence of the operator is verified by applying Volkov theorem, and its convergence rate is estimated. Those re-sults further promote some of the conclusions of (p, q)-Bernstein operator.

1. 引言

q微积分在逼近中的发展推动了(p, q)微分学步入逼近理论。Mursaleen于2015年首次在q-Bernstein算子的基础上提出(p, q)-Bernstein算子 [8]，实现了q-Bernstein算子性质的推广。自此，有关于(p, q)型算子呈现于世人面前。2016年，Acar在文献 [9] 中构建了两元(p, q)-Bernstein-Kantorovich算子并证明该算子一些的逼近结论。由此可知，关于(p, q)型二元算子逼近问题的研究正在持续发展中。本文构建出二元(p, q)Bernstein算子，证明算子的一些逼近相关的定理，从而更进一步推广一元算子的逼近性质，更加丰富逼近理论的完整性。

2. 知识储备

${\left[n\right]}_{p,q}=\frac{{p}^{n}-{q}^{n}}{p-q},\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }n=0,1,2,\cdots$

${\left[n\right]}_{p,q}!=\left\{\begin{array}{l}1,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }n=0;\\ {\left[n\right]}_{p,q}{\left[n-1\right]}_{p,q}\cdots \left[1\right],\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }n=1,2,\cdots \end{array}$

${\left[\begin{array}{l}n\\ k\end{array}\right]}_{p,q}=\frac{{\left[n\right]}_{p,q}!}{{\left[k\right]}_{p,q}!{\left[n-k\right]}_{p,q}!}$

${B}_{n}^{p,q}\left(f;x\right)=\underset{k=0}{\overset{n}{\sum }}{b}_{n,k}^{p,q}\left(x\right)f\left(\frac{{p}^{n-k}{\left[k\right]}_{p,q}}{{\left[n\right]}_{p,q}}\right),$

${B}_{{n}_{1}{n}_{2}}\left(f;x,y\right)=\frac{1}{{p}_{1}^{\frac{{n}_{1}\left({n}_{1}-1\right)}{2}}}\frac{1}{{p}_{2}{}^{\frac{{n}_{2}\left({n}_{2}-1\right)}{2}}}\underset{{k}_{1}=0}{\overset{{n}_{1}}{\sum }}\underset{{k}_{2}=0}{\overset{{n}_{2}}{\sum }}{b}_{{n}_{1}{n}_{2}{k}_{1}{k}_{2}}\left(x,y\right)\left(x,y\right)f\left(\frac{{p}_{1}^{{n}_{1}-{k}_{1}}{\left[{k}_{1}\right]}_{{p}_{1},{q}_{1}}}{{\left[{n}_{1}\right]}_{{p}_{1},{q}_{1}}},\frac{{p}_{2}^{{n}_{2}-{k}_{2}}{\left[{k}_{2}\right]}_{{p}_{2},{q}_{2}}}{{\left[{n}_{2}\right]}_{{p}_{2},{q}_{2}}}\right)$

${b}_{{n}_{1}{n}_{2}{k}_{1}{k}_{2}}\left(x,y\right)={\left[\begin{array}{c}{n}_{1}\\ {k}_{1}\end{array}\right]}_{{p}_{1},{q}_{1}}{\left[\begin{array}{c}{n}_{2}\\ {k}_{2}\end{array}\right]}_{{p}_{2},{q}_{2}}{p}_{1}^{\frac{{k}_{1}\left({k}_{1}-1\right)}{2}}{p}_{2}^{\frac{{k}_{2}\left({k}_{2}-1\right)}{2}}{x}^{{k}_{1}}{y}^{{k}_{2}}\underset{s=0}{\overset{{n}_{1}-{k}_{1}-1}{\prod }}\left({p}_{1}^{s}-{q}_{1}^{s}x\right)\underset{s=0}{\overset{{n}_{2}-{k}_{2}-1}{\prod }}\left({p}_{2}^{s}-{q}_{2}^{s}y\right)\text{ }\text{ }.$

$\begin{array}{l}{B}_{n}^{p,q}\left({e}_{0};x\right)=1,\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }{B}_{n}^{p,q}\left({e}_{1};x\right)=x,\\ {B}_{n}^{p,q}\left({e}_{2};x\right)=\frac{{p}^{n-1}}{{\left[n\right]}_{p,q}}x+\frac{q{\left[n-1\right]}_{p,q}}{{\left[n\right]}_{p,q}}{x}^{2}.\end{array}$

$\begin{array}{l}{B}_{{n}_{1}{n}_{2}}\left({e}_{00};x,y\right)={e}_{00}\left(x,y\right);\\ {B}_{{}_{{n}_{1}{n}_{2}}}\left({e}_{10};x,y\right)={e}_{10}\left(x,y\right);\\ {B}_{{n}_{1}{n}_{2}}\left({e}_{01};x,y\right)={e}_{01}\left(x,y\right);\\ {B}_{{n}_{1}{n}_{2}}\left({e}_{11};x,y\right)={e}_{11}\left(x,y\right);\\ {B}_{{n}_{1}{n}_{2}}\left({e}_{20};x,y\right)={e}_{20}\left(x,y\right)+\frac{{p}_{1}^{{n}_{1}-1}x\left(1-x\right)}{{\left[{n}_{1}\right]}_{{p}_{1},{q}_{1}}};\\ {e}_{{n}_{1}{n}_{2}}\left({e}_{02};x,y\right)={e}_{02}\left(x,y\right)+\frac{{p}_{2}^{{n}_{2}-1}y\left(1-y\right)}{{\left[{n}_{2}\right]}_{{p}_{2},{q}_{2}}}.\end{array}$

$\begin{array}{l}{B}_{{n}_{1}}{}_{{n}_{2}}\left({e}_{00};x,y\right)\\ =\frac{1}{{p}_{1}{}^{\frac{{n}_{1}\left({n}_{1}-1\right)}{2}}}\frac{1}{{p}^{\frac{{n}_{2}\left({n}_{2}-1\right)}{2}}}\underset{{k}_{1}=0}{\overset{{n}_{1}}{\sum }}\underset{{k}_{2}=0}{\overset{{n}_{2}}{\sum }}{\left[\begin{array}{c}{n}_{1}\\ {k}_{1}\end{array}\right]}_{{p}_{1},{q}_{1}}{\left[\begin{array}{c}{n}_{2}\\ {k}_{2}\end{array}\right]}_{{p}_{2},{q}_{2}}{p}_{1}^{\frac{{k}_{1}\left({k}_{1}-1\right)}{2}}{p}_{2}^{\frac{{k}_{2}\left({k}_{2}-1\right)}{2}}{x}^{{k}_{1}}{y}^{{k}_{2}}×\underset{s=0}{\overset{{n}_{1}-{k}_{1}-1}{\prod }}\left({p}_{1}^{s}-{q}_{1}^{s}x\right)\underset{s=0}{\overset{{n}_{2}-{k}_{2}-1}{\prod }}\left({p}_{2}^{s}-{q}_{2}^{s}y\right)\\ =\frac{1}{{p}_{1}{}^{\frac{{n}_{1}\left({n}_{1}-1\right)}{2}}}\underset{{k}_{1}=0}{\overset{{n}_{1}}{\sum }}{\left[\begin{array}{c}{n}_{1}\\ {k}_{1}\end{array}\right]}_{{p}_{1},{q}_{1}}{p}^{\frac{{k}_{1}\left({k}_{1}-1\right)}{2}}{x}^{{k}_{1}}\underset{s=0}{\overset{{n}_{1}-{k}_{1}-1}{\prod }}\left({p}_{1}^{s}-{q}_{1}^{s}x\right)×\frac{1}{{p}_{2}{}^{\frac{{n}_{2}\left({n}_{2}-1\right)}{2}}}{\underset{{k}_{2}=0}{\overset{{n}_{2}}{\sum }}\left[\begin{array}{c}{n}_{2}\\ {k}_{2}\end{array}\right]}_{{p}_{2},{q}_{2}}{p}^{\frac{{k}_{2}\left({k}_{2}-1\right)}{2}}{y}^{{k}_{2}}\underset{s=0}{\overset{{n}_{2}-{k}_{2}-1}{\prod }}\left({p}_{2}^{s}-{q}_{2}^{s}y\right)\\ ={B}_{{n}_{1}}\left({e}_{0};x\right){B}_{{n}_{2}}\left({e}_{0};y\right)={e}_{00}\left(x,y\right),\end{array}$

$\begin{array}{l}{B}_{{n}_{1}}{}_{{n}_{2}}\left({e}_{10};x,y\right)\\ =\frac{1}{{p}^{\frac{{n}_{1}\left({n}_{1}-1\right)}{2}}}\frac{1}{{p}^{\frac{{n}_{2}\left({n}_{2}-1\right)}{2}}}\underset{{k}_{1}=0}{\overset{{n}_{1}}{\sum }}\underset{{k}_{2}=0}{\overset{{n}_{2}}{\sum }}{\left[\begin{array}{c}{n}_{1}\\ {k}_{1}\end{array}\right]}_{{p}_{1},{q}_{1}}{\left[\begin{array}{c}{n}_{2}\\ {k}_{2}\end{array}\right]}_{{p}_{2},{q}_{2}}{p}^{\frac{{k}_{1}\left({k}_{1}-1\right)}{2}}{p}^{\frac{{k}_{2}\left({k}_{2}-1\right)}{2}}{x}^{{k}_{1}}{y}^{{k}_{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }×\underset{s=0}{\overset{{n}_{1}-{k}_{1}-1}{\prod }}\left({p}_{1}^{s}-{q}_{1}^{s}x\right)\underset{s=0}{\overset{{n}_{2}-{k}_{2}-1}{\prod }}\left({p}_{2}^{s}-{q}_{2}^{s}y\right)\frac{{\left[{k}_{1}\right]}_{{p}_{1},{q}_{1}}}{{p}_{1}^{{k}_{1}-{n}_{1}}{\left[{n}_{1}\right]}_{{p}_{1},{q}_{1}}}\\ =\frac{1}{{p}^{\frac{{n}_{1}\left({n}_{1}-1\right)}{2}}}\frac{1}{{p}^{\frac{{n}_{2}\left({n}_{2}-1\right)}{2}}}\underset{{k}_{1}=0}{\overset{{n}_{1}}{\sum }}{\left[\begin{array}{c}{n}_{1}\\ {k}_{1}\end{array}\right]}_{{p}_{1},{q}_{1}}{p}_{1}^{\frac{{k}_{1}\left({k}_{1}-1\right)}{2}}{x}^{{k}_{1}}\underset{s=0}{\overset{{n}_{1}-{k}_{1}-1}{\prod }}\left({p}_{1}^{s}-{q}_{1}^{s}x\right)\frac{{\left[{k}_{1}\right]}_{{p}_{1},{q}_{1}}}{{p}_{1}^{{k}_{1}-{n}_{1}}{\left[{n}_{1}\right]}_{{p}_{1},{q}_{1}}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }×{\underset{{k}_{2}=0}{\overset{{n}_{2}}{\sum }}\left[\begin{array}{c}{n}_{2}\\ {k}_{2}\end{array}\right]}_{{p}_{2},{q}_{2}}{p}^{\frac{{k}_{2}\left({k}_{2}-1\right)}{2}}{y}^{{k}_{2}}\underset{s=0}{\overset{{n}_{2}-{k}_{2}-1}{\prod }}\left({p}_{2}^{s}-{q}_{2}^{s}y\right)\\ ={B}_{{n}_{1}}\left({e}_{1};x\right){B}_{{n}_{2}}\left({e}_{0},y\right)={e}_{10}\left(x,y\right)\end{array}$

$\begin{array}{l}{B}_{{n}_{1}}{}_{{n}_{2}}\left({e}_{11};x,y\right)\\ =\frac{1}{{p}^{\frac{{n}_{1}\left({n}_{1}-1\right)}{2}}}\frac{1}{{p}^{\frac{{n}_{2}\left({n}_{2}-1\right)}{2}}}{\underset{{k}_{1}=0}{\overset{{n}_{1}}{\sum }}\underset{{k}_{2}=0}{\overset{{n}_{2}}{\sum }}\left[\begin{array}{c}{n}_{1}\\ {k}_{1}\end{array}\right]}_{{p}_{1},{q}_{1}}{\left[\begin{array}{c}{n}_{2}\\ {k}_{2}\end{array}\right]}_{{p}_{2},{q}_{2}}{p}_{1}^{\frac{{k}_{1}\left({k}_{1}-1\right)}{2}}{p}_{2}^{\frac{{k}_{2}\left({k}_{2}-1\right)}{2}}{x}^{{k}_{1}}{y}^{{k}_{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }×\underset{s=0}{\overset{{n}_{1}-{k}_{1}-1}{\prod }}\left({p}_{1}^{s}-{q}_{1}^{s}x\right)\underset{s=0}{\overset{{n}_{2}-{k}_{2}-1}{\prod }}\left({p}_{2}^{s}-{q}_{2}^{s}y\right)\frac{{\left[{k}_{1}\right]}_{{p}_{1},{q}_{1}}}{{p}_{1}^{{k}_{1}-{n}_{1}}{\left[{n}_{1}\right]}_{{p}_{1},{q}_{1}}}\frac{{\left[{k}_{2}\right]}_{{p}_{2},{q}_{2}}}{{p}_{2}^{{k}_{2}-{n}_{2}}{\left[{n}_{2}\right]}_{{p}_{2},{q}_{2}}}\\ =\frac{1}{{p}_{1}{}^{\frac{{n}_{1}\left({n}_{1}-1\right)}{2}}}{\underset{{k}_{1}=0}{\overset{{n}_{1}}{\sum }}\left[\begin{array}{c}{n}_{1}\\ {k}_{1}\end{array}\right]}_{{p}_{1},{q}_{1}}{p}_{1}^{\frac{{k}_{1}\left({k}_{1}-1\right)}{2}}{x}^{{k}_{1}}\underset{s=0}{\overset{{n}_{1}-{k}_{1}-1}{\prod }}\left({p}_{1}^{s}-{q}_{1}^{s}x\right)\frac{{\left[{k}_{1}\right]}_{{p}_{1},{q}_{1}}}{{p}_{1}^{{k}_{1}-{n}_{1}}{\left[{n}_{1}\right]}_{{p}_{1},{q}_{1}}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }×\frac{1}{{p}_{1}{}^{\frac{{n}_{2}\left({n}_{2}-1\right)}{2}}}\underset{{k}_{2}=0}{\overset{{n}_{2}}{\sum }}{\left[\begin{array}{c}{n}_{2}\\ {k}_{2}\end{array}\right]}_{{p}_{2},{q}_{2}}{p}_{2}^{\frac{{k}_{2}\left({k}_{2}-1\right)}{2}}{y}^{{k}_{2}}\underset{s=0}{\overset{{n}_{2}-{k}_{2}-1}{\prod }}\left({p}_{2}^{s}-{q}_{2}^{s}y\right)\frac{{\left[{k}_{2}\right]}_{{p}_{2},{q}_{2}}}{{p}_{2}^{{k}_{2}-{n}_{2}}{\left[{n}_{2}\right]}_{{p}_{2},{q}_{2}}}\\ ={B}_{{n}_{1}}\left({e}_{1},x\right){B}_{{n}_{2}}\left({e}_{1},y\right)={e}_{11}\left(x,y\right)\end{array}$

$\begin{array}{l}{B}_{{n}_{1}}{}_{{n}_{2}}\left({e}_{02};x,y\right)\\ =\frac{1}{{p}_{1}^{\frac{{n}_{1}\left({n}_{1}-1\right)}{2}}}\frac{1}{{p}_{2}^{\frac{{n}_{2}\left({n}_{2}-1\right)}{2}}}\underset{{k}_{1}=0}{\overset{{n}_{1}}{\sum }}\underset{{k}_{2}=0}{\overset{{n}_{2}}{\sum }}{\left[\begin{array}{c}{n}_{1}\\ {k}_{1}\end{array}\right]}_{{p}_{1},{q}_{1}}{\left[\begin{array}{c}{n}_{2}\\ {k}_{2}\end{array}\right]}_{{p}_{2},{q}_{2}}{p}_{1}^{\frac{{k}_{1}\left({k}_{1}-1\right)}{2}}{p}_{2}^{\frac{{k}_{2}\left({k}_{2}-1\right)}{2}}{x}^{{k}_{1}}{y}^{{k}_{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }×\underset{s=0}{\overset{{n}_{1}-{k}_{1}-1}{\prod }}\left({p}_{1}^{s}-{q}_{1}^{s}x\right)\underset{s=0}{\overset{{n}_{2}-{k}_{2}-1}{\prod }}\left({p}_{2}^{s}-{q}_{2}^{s}y\right)\frac{{\left[{k}_{2}\right]}_{{p}_{2,}{q}_{2}}^{2}}{{p}_{2}^{2{k}_{2}-2{n}_{2}}{\left[{n}_{2}\right]}_{{p}_{2},{q}_{2}}^{2}}\\ ={y}^{2}+\frac{\left({q}_{2}{\left[{n}_{2}-1\right]}_{{p}_{2},{q}_{2}}-{\left[{n}_{2}\right]}_{{p}_{2},{q}_{2}}\right){y}^{2}+{p}_{2}^{{n}_{2}-1}y}{{\left[{n}_{2}\right]}_{{p}_{2},{q}_{2}}}\\ ={y}^{2}+\frac{{p}_{2}^{{n}_{2}-1}y\left(1-y\right)}{{\left[{n}_{2}\right]}_{{p}_{2},{q}_{2}}}={B}_{{n}_{1}}\left({e}_{0},x\right){B}_{{n}_{2}}\left({e}_{2},y\right)\\ ={e}_{02}\left(x,y\right)+\frac{{p}_{2}^{{n}_{2}-1}y\left(1-y\right)}{{\left[{n}_{2}\right]}_{{p}_{2},{q}_{2}}}.\end{array}$

$\begin{array}{l}{B}_{{n}_{1}{n}_{2}}\left({e}_{10}-x;x,y\right)=0;\\ {B}_{{n}_{1}{n}_{2}}\left({e}_{01}-x;x,y\right)=0;\\ {B}_{{n}_{1}{n}_{2}}\left({\left({e}_{10}-x\right)}^{2};x,y\right)=\frac{{p}_{1}^{{n}_{1}-1}x\left(1-x\right)}{{\left[{n}_{1}\right]}_{{p}_{1},{q}_{1}}};\\ {B}_{{n}_{1}{n}_{2}}\left({\left({e}_{01}-x\right)}^{2};x,y\right)=\frac{{p}_{2}^{{n}_{2}-1}y\left(1-y\right)}{{\left[{n}_{2}\right]}_{{p}_{2},{q}_{2}}}.\end{array}$

3. 主要结果

$\omega \left(f:{\delta }_{1},{\delta }_{2}\right)=\mathrm{sup}\left\{|f\left(t,s\right)-f\left(x,y\right)|:\left(t,s\right)\left(x,y\right)\in \left({I}_{1}×{I}_{2}\right),|t-x|\le {\delta }_{1},|s-y|\le {\delta }_{2}\right\}$

$\begin{array}{l}\left(\text{i}\right)\text{\hspace{0.17em}}\text{ }\omega \left(f:{\delta }_{1},{\delta }_{2}\right)\to 0,若{\delta }_{1},{\delta }_{2}\to \text{0}\\ \left(\text{ii}\right)\text{\hspace{0.17em}}f\left(t,s\right)-f\left(x,y\right)\le \omega \left(f:{\delta }_{1},{\delta }_{2}\right)\left(1+\frac{|t-x|}{{\delta }_{1}}\right)\left(1+\frac{|s-y|}{{\delta }_{2}}\right)\end{array}$

${\alpha }_{1},{\alpha }_{2}$ 阶Lipschitz条件的二元函数f：对于 $\forall \left(t,s\right),\left(x,y\right)\in {I}^{2}$$f\in C\left({I}^{2}\right)$$0<{\alpha }_{1}\le 1,0<{\alpha }_{2}\le 1$，则存在常数 $M>0$，使得 $|f\left(t,s\right)-f\left(x,y\right)|\le M{|t-x|}^{{\alpha }_{1}}{|s-y|}^{{\alpha }_{2}}$ ；记为 $f\in Li{p}_{M}\left({\alpha }_{1},{\alpha }_{2}\right)$

$\begin{array}{l}‖{B}_{{n}_{1}{n}_{2}}\left({e}_{00}:x,y\right)-{e}_{00}‖=0;\\ ‖{B}_{{n}_{1}{n}_{2}}\left({e}_{10}:x,y\right)-{e}_{10}‖=0;\\ ‖{B}_{{n}_{1}{n}_{2}}\left({e}_{01}:x,y\right)-{e}_{01}‖=0;\end{array}$

，即成立。

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