Approximation for modification of exponential type operators connected with $$x(x+1)^2$$

Approximation by exponential type summation integral operator having weight of α-Baskakov basis function

“…To explore simultaneous approximation for the operators, the following lemma is required: Lemma 3.1. [9] There exist the polynomials q i,j,r (x) independent of n and k such that…”

“…In [8], Gupta and Agarwal observed that the Durrmeyer variant of Ismail and May operators can not be defined due to non-convergence behaviour of weight function r n,j (y) under the integral sign. Recently, they defined the Durremeyer variant of the operator (1.1) by using the weight of Baskakov basis function [9].…”

Estimation using a summation integral operator of exponential type with a weight derived from the α‐Baskakov basis function

“…Gupta and Agarwal [15] observed that Durrmeyer‐type modification of Ismail and May operators cannot be defined due to nonconvergence behaviour of $$$ {r}_{m,i}(y) $$$ under the integral sign. Recently, they defined the Durremeyer variant of the operator () by using the weight of the Baskakov basis function [16]. Recently, Aral and Erbay [17] proposed a modification of the Baskakov operator depending upon a nonnegative real parameter $$$ \alpha $$$.…”

Ismail-May-Kantorovich operators preserving affine functions

“…These operators were proposed in [18, (3.14)] while the authors constructed several exponential type operators. The approximation properties of these operators considering different basis functions have been studied by the authors in [13] and [14].…”