Sz\xe1sz-Durrmeyer operators involving Boas-Buck polynomials of blending type

Manjari Sidharth,P N Agrawal,Serkan Araci

doi:10.1186/s13660-017-1396-x

Manjari Sidharth, P N Agrawal + Show 1 more

Open Access

https://doi.org/10.1186/s13660-017-1396-x

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Abstract

The present paper introduces the Szász-Durrmeyer type operators based on Boas-Buck type polynomials which include Brenke type polynomials, Sheffer polynomials and Appell polynomials considered by Sucu et al. (Abstr. Appl. Anal. 2012:680340, 2012). We establish the moments of the operator and a Voronvskaja type asymptotic theorem and then proceed to studying the convergence of the operators with the help of Lipschitz type space and weighted modulus of continuity. Next, we obtain a direct approximation theorem with the aid of unified Ditzian-Totik modulus of smoothness. Furthermore, we study the approximation of functions whose derivatives are locally of bounded variation.

Highlights

For a real-valued bounded function f on [, ], Bernstein [ ] defined a sequence of polynomials given by n r Bn(f ; x) = f n r=n xr( – x)n–r, ∀x ∈ [, ] and n ∈ N, r to provide a very simple and elegant proof of the Weierstrass approximation theorem
Motivated by the above work, in the present paper we define Szász-Durrmeyer type operators based on Boas-Buck type polynomials as follows
3.2 Unified modulus theorem We investigate a direct approximation theorem by utilizing the unified Ditzian-Totik modulus of smoothness ωφτ (f, t), ≤ τ ≤

Summary

Introduction

For a real-valued bounded function f on [ , ], Bernstein [ ] defined a sequence of polynomials given by n r. In [ ], Sucu et al introduced the Szász operators involving Boas-Buck type polynomials as follows: Bn(f ; x) := A( )G(nxH( )) pk(nx)f k , n x ≥ , n ∈ N,. Motivated by the above work, in the present paper we define Szász-Durrmeyer type operators based on Boas-Buck type polynomials as follows. We obtain the moments for the operators defined by ), we make the following assumptions on the analytic functions A(t), H(t) and G(t). It is to be noted that the following assumptions are valid pointwise These assumptions will be needed to prove Theorems , and of this paper which are pointwise results. As a result of the above assumptions, applying Lemma , we reach the following important result

Results and discussion

Weighted approximation Let

Conclusion