Approximation Theorems for Generalized Complex Kantorovich‐Type Operators

Abstract

The order of simultaneous approximation and Voronovskaja-type results with quantitative estimate for complex q-Kantorovich polynomials q > 0 attached to analytic functions on compact disks are obtained. In particular, it is proved that for functions analytic in {z ∈ C : |z| < R}, R > q, the rate of approximation by the q-Kantorovich operators q > 1 is of order q −n versus 1/n for the classical Kantorovich operators.

Certain Identities Associated with (p,q)-Binomial Coefficients and (p,q)-Stirling Polynomials of the Second Kind

“…Even though a generalization of Bernstein polynomials associated with q-integers was suggested in 1987 (see ([60], Section 1)), the q-analogue of Bernstein polynomials (23) have been received as a standard definition and investigated in such diverse ways as (an extension of several variables [61]; other q-polynomials and operators [7,20,[62][63][64][65]; other types of Bernstein polynomials [19,66]; convergence and iterates [9,60,67]; monotonicity [48]; Cauchy kernel [68]; norm estimates [69]; unbounded function [70]; overview of the first decade [71]).…”

“…We begin by noting that the vanishing identity (66) follows immediately from either (65) or (63) when, respectively, k = 0 and = 0. Let L p,q (ν; k) be the left-hand member of (65). We will use induction on k to prove (65).…”

“…Let L p,q (ν; k) be the left-hand member of (65). We will use induction on k to prove (65). The case L p,q (ν; 0) = 0 follows from (63) when = 0.…”

Bernstein-Kantorovich operators on multidimensional cube

“…Properties of convergence and Voronovskaja-type results for multidimensional Bernstein-Kantorovich operators will be investigated in our paper. Properties of convergence and Voronovskaja-type results of linear and positive operators have been studied by many authors in unidimensional case, see [2], [4], [11], [12], [13], [14], [15].…”

Approximation by Genuineq-Bernstein-Durrmeyer Polynomials in Compact Disks in the Caseq>1

“…Also, for the case > 1, exact quantitative estimates and quantitative Voronovskaja-type results for complex -Lorentz polynomials, -Stancu polynomials [20], -Stancu-Faber polynomials, -Bernstein-Faber polynomials, -Kantorovich polynomials [21], -Szász-Mirakjan operators [22] obtained by different researchers are collected in the recent book of Gal [23]. In this book the definition and study of complex -Durrmeyer-kind operators for > 1 presented an open problem.…”