Approximation by operators including generalized Appell polynomials

Abstract

In this work, the problem of the approximation by certain polynomials is addressed. A new type operators sequence including generalized Appell polynomials are defined, qualitative and quantitative approximation theorems are proved. Some explicit examples of our operators involving Hermite polynomials of ν variance, Gould-Hopper polynomials and Miller-Lee polynomials are given. Also, we present some numerical examples to confirm our theoretical results.

Baskakov‐Durrmeyer type operators involving generalized Appell Polynomials

“…The values of the moments M n (v i ; y) for i = 0, 1, 2 are given in 6 , while the values of M n (v i ; y), for i = 3, 4 have been obtained by us by simple calculations and hence we skip the details. Lemma 2.…”

“…İçöz et al defined a positive linear operator using generalized Appell polynomials as where it is assumed that (i) A ( w (1))≠0, w ′ (1)=1, p k ( y ) ≥ 0, k =0,1,2,…, (ii) (iii) and power series converge for | v |< R , R >1). …”

“…Içöz et al 6 defined a positive linear operator using generalized Appell polynomials as M n ( ; ) = 1 A(w(1))B(n w (1)…”

“…where it is assumed that (i) A(w(1)) ≠ 0, w ′ (1) = 1, p k (y) ≥ 0, k = 0, 1, 2, … , (ii) B ∶ R → (0, ∞), (iii) (6) and power series (7) converge for |v| < R, R > 1).…”

“…For B(v) = e v , A(v) = 1 and w(v) = 1, using relation (6), these operators include the Szász-Baskakov type operators given by (4). The aim of the paper is to investigate uniform convergence theorem, error estimates by means of moduli of continuity, Voronovskaya-type asymptotic theorem, quantitative-Voronovskaya result, Grüss Voronovskaya-type theorem and an error estimate for functions with derivatives of bounded variation.…”