A new kind of variant of the Kantorovich type modification operators introduced by D. D. Stancu

Abstract

In the present article we investigate a variant of the Kantorovich type modification defined by Kajla (2018) i.e. we introduce a function ζ(ϰ)in the operators defined by Kajla (2018) s.t. ζ(ϰ)is infinitely differentiable function on [0,1],ζ(0)=0,ζ(1)=1and ζ′(ϰ)>0,∀ϰ∈[0,1]. We prove an approximation theorem with the help of Bohman–Korovkin’s theorem, and investigate the estimate of the rate of convergence by means of modulus of smoothness and Lipschitz type function for these operators and the approximation of functions with derivatives of bounded variation are also studied. We study an approximation theorem with the help of Bohman–Korovkin’s principle in A−statistical convergence. Finally, by means of a numerical example we illustrate the convergence of these operators to certain functions through graphs with the help of MATHEMATICA and show that a careful choice of the function ζ(ϰ)leads to a better approximation than the Kantorovich type modification defined by Kajla (2018).

Keywords

• Modulus of continuity
• Lipschitz type maximal function
• Peetre’s K-functional
• Asymptotic formula
• Local approximation