Inequalities for a class of multivariate operators

Jianwei Zhao

doi:10.1186/1029-242x-2012-175

Abstract

This paper introduces and studies a class of generalized multivariate Bernstein operators defined on the simplex. By means of the modulus of continuity and so-called Ditzian-Totik’s modulus of function, the direct and inverse inequalities for the operators approximating multivariate continuous functions are simultaneously established. From these inequalities, the characterization of approximation of the operators follows. The obtained results include the corresponding ones of the classical Bernstein operators. MSC:41A25, 41A36, 41A60, 41A63.

Highlights

1 Introduction Let N be the set of natural numbers, and {sn}∞ n= be a sequence
When sn =, Lnf reduce to the classical Bernstein operators, Bnf, given by nk (Bnf )(x) := f n Pn,k(x)
We will introduce and study the multivariate version defined on the simplex of the generalized Bernstein operators given by ( )

Summary

Introduction

Let N be the set of natural numbers, and {sn}∞ n= (sn ≥ , sn ∈ N) be a sequence. In [ ], Cao introduced the following generalized Bernstein operators defined on [ , ]: n sn–k+j (Lnf )(x) := sn k= f j=n + sn – Pn,k (x), ( )where x ∈ [ , ], f ∈ C[ , ], and Pn,k(x) :=n k xk( – x)n–k.Clearly, when sn = , Lnf reduce to the classical Bernstein operators, Bnf , given by nk (Bnf )(x) := f n Pn,k(x). k=. In [ ], Cao introduced the following generalized Bernstein operators defined on [ , ]: When sn = , Lnf reduce to the classical Bernstein operators, Bnf , given by nk (Bnf )(x) := f n Pn,k(x). Cao [ ] proved that the necessary and sufficient condition of convergence for the operators is lim√n→∞(sn/n) = , and he proved that for n ∈ Q = {n : n ∈ N, and < (sn – )/n + / n ≤ } the following estimate of approximation degree holds: Lnf – f

Results

Conclusion