General Summability Methods in the Approximation by Bernstein–Chlodovsky Operators

Abstract

In this paper, by using regular summability methods we modify the Bernstein–Chlodovsky operators in order get more general and powerful results than the classical aspects. We study Korovkin-type approximation theory on weighted spaces. As a special case, it is possible to Cesàro approximate (arithmetic mean convergence) to the test function $e_2(x)=x^2$ although it fails for the classical Bernstein–Chlodovsky operators. At the end of the paper, we extend our results to the multi-dimensional case.

KEYWORDS AND PHRASES:

• Bernstein–Chlodovsky operators
• Cesàro summability
• modulus of continuity
• summability methods

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 41A36
• 41A25 (primary)
• and 40A05
• 40A25 (secondary)