Rate of convergence for Szasz–Mirakyan–Durrmeyer operators with derivatives of bounded variation

Abstract

In the present paper, we estimate the rate of convergence for Szasz–Mirakyan–Durrmeyer operators with derivatives of bounded variation.

Introduction

Let DB_γ(0, ∞),(γ ⩾ 0) be the class of all locally integrable functions defined on (0, ∞), with the growth condition ∣f(t)∣ ⩽ Me^γt, M > 0 and f′ ∈ BV on every finite subinterval [0, ∞). Then for a function f ∈ DB_γ(0, ∞).

We consider the Szasz–Mirakyan–Durrmeyer operators [4], which are defined by

$$S_n(f\text{,}x)=n\sum _{k=1}^{\infty }{p}_{n\text{,}k}(x){\int }_0^{\infty }{p}_{n\text{,}k}(t)f(t)\mathrm{d}t\text{,}x\in [0\text{,}\infty )\text{,}$$

where

$$p_{n\text{,}k}(x)={\mathrm{e}}^{-\mathit{nx}}\frac{(\mathit{nx}{)}^k}{k!}\text{.}$$

Alternately we may rewrite (1) as

$$S_n(f\text{,}x)={\int }_0^{\infty }{k}_n(x\text{,}t)f(t)\mathrm{d}t\text{,}$$

where

$$k_n(x\text{,}t)=n\sum _{k=0}^{\infty }{p}_{n\text{,}k}(x)p_{n\text{,}k}(t)\text{.}$$

Also let ${\beta }_n(x\text{,}t)={\int }_0^t{k}_n(x\text{,}s)\mathrm{d}s$, then it is easily verified that

$${\beta }_n(x\text{,}\infty )={\int }_0^{\infty }{k}_n(x\text{,}s)\mathrm{d}s=1\text{.}$$

Very recently Srivastava et al. [5] and Gupta et al. [2] estimated the convergence rates for functions having derivatives of bounded variation for certain integral operators and second kind of beta operators respectively. Also very recently Gupta et al. [3] studied some other summation-integral type operators. In the present paper, we extend the results of Gupta et al. [2] and study the rate of convergence by means of the decomposition technique of functions having derivatives of bounded variation. More precisely the functions having derivatives of bounded variation on every finite subinterval on the interval [0, ∞) be defined as

$$f(x)=f(0)+{\int }_0^x\psi (t)\mathrm{d}t\text{,}0<a⩽x⩽b\text{,}$$

where ψ is a function of bounded variation on [a, b]. We denote the auxiliary function f_x, by

$$f_x(t)=\left\{\begin{array}{c}f(t)-f(x^{-})\text{,}0⩽t<x\text{,}\hfill \\ 0\text{,}t=x\text{,}\hfill \\ f(t)-f(x^{+})\text{,}x<t<\infty .\hfill \end{array}\right.$$