Bivariate Szász-Type Operators Based on Multiple Appell Polynomials

Abstract

We introduce bivariate case of the Szász-type operators based on multiple Appell polynomials introduced by Varma (Stud. Univ. Babeş -Bolyai Math. 58, 361–369 (2013)). We establish a uniform convergence theorem and determine the degree of approximation in terms of the partial moduli of continuity of the approximated function. We estimate the error in simultaneous approximation of the function by the bivariate operators by using finite differences. We investigate the degree of approximation of the bivariate operators by means of the Peetre’s K-functional. The rate of convergence of these operators is determined for twice continuously differentiable functions by Voronovskaja-type asymptotic theorem. The weighted approximation properties are derived for unbounded functions with a polynomial growth. Lastly, we introduce the associated generalized boolean sum (GBS) of the bivariate operators to study the approximation of Bögel-continuous and Bögel-differentiable functions and establish the approximation degree with the aid of the Lipschitz class of Bögel-continuous functions and the mixed modulus of smoothness.