Research Article

POINTWISE APPROXIMATION BY BERNSTEIN POLYNOMIALS

GANCHO TACHEVa1

a1 Department of Mathematics, University of Architecture, Civil Engineering and Geodesy, BG-1046, Sofia, Bulgaria (email: gtt_fte@uacg.acad.bg)

Abstract

We improve the degree of pointwise approximation of continuous functions f(x) by Bernstein operators, when x is close to the endpoints of [0,1]. We apply the new estimate to establish upper and lower pointwise estimates for the test function g(x)=xlog (x)+(1−x)log (1−x). At the end we prove a general statement for pointwise approximation by Bernstein operators.

(Received April 22 2011)

2010 Mathematics subject classification

  • primary 41A10; secondary 41A15;
  • 41A25;
  • 41A36

Keywords and phrases

  • Bernstein polynomials;
  • Direct theorems;
  • Ditzian–Totik moduli of smoothness