Equivalent theorem of approximation by linear combination of weighted Baskakov\u2013Kantorovich operators in Orlicz spaces

Ling-Xiong Han,Bai-Ni Guo,Feng Qi

doi:10.1186/s13660-019-2174-8

Abstract

In this paper, we introduce the Orlicz space corresponding to the Young function and, by virtue of the equivalent theorem between the modified K-functional and modulus of smoothness, establish the direct, inverse, and equivalent theorems for linear combination of the Jacobi weighted Baskakov–Kantorovich operators in the Orlicz spaces.

Highlights

For proceeding smoothly, we recall from [22] some definitions and related results
Kumar and Acar [14] introduced a modification of generalized Baskakov–Durrmeyer operators of the Stancu type and studied their approximation properties
Goyal and Agrawal [8] introduced the Bézier variant of the generalized Baskakov–Kantorovich operators, established a direct approximation theorem with the aid of the Ditzian–Totik modulus of smoothness, and studied the rate of convergence for the functions having the derivatives of bounded variation for these operators

Summary

To approximate functions in the

There are many approximation results on operators of the Baskakov type in the space C[0, ∞) or Lp[0, ∞). Kumar and Acar [14] introduced a modification of generalized Baskakov–Durrmeyer operators of the Stancu type and studied their approximation properties. Gadjev [7] studied the approximation of functions by the Baskakov–Kantorovich operator in the space Lp[0, ∞) and obtained the double inequality. In [11], we obtained approximation properties for linear combinations of modified summation operators of integral type in the Orlicz space. Basing on these conclusions, we discover in this paper approximation properties for linear combinations of the Baskakov– Kantorovich operators. Our main results in this paper can be stated as the following three theorems

Let f

We now estimate

For x