Approximation properties of generalized Baskakov operators

Abstract

The present article is a continuation of the work done by Aral and Erbay [1]. We discuss the rate of convergence of the generalized Baskakov operators considered in the above paper with the aid of the second order modulus of continuity and the unified Ditzian Totik modulus of smoothness. A bivariate case of these operators is also defined and the degree of approximation by means of the partial and total moduli of continuity and the Peetre's K-functional is studied. A Voronovskaya type asymptotic result is also established. Further, we construct the associated Generalized Boolean Sum (GBS) operators and investigate the order of convergence with the help of mixed modulus of smoothness for the Bögel continuous and Bögel differentiable functions. Some numerical results to illustrate the convergence of the above generalized Baskakov operators and its comparison with the GBS operators are also given using Matlab algorithm.

Keywords

• voronovskaya theorem
• ditzian-totik modulus of smoothness
• peetre's k-functional
• bögel continuous function
• bögel differentiable function