Modified Stancu operators based on inverse Polya Eggenberger distribution

Abstract

Abstract In this paper, we construct a sequence of modified Stancu-Baskakov operators for a real valued function bounded on [ 0 , ∞ ) $[0,\infty)$ , based on a function τ ( x ) $\tau(x)$ . This function τ ( x ) $\tau(x)$ is infinite times continuously differentiable on [ 0 , ∞ ) $[0,\infty)$ and satisfy the conditions τ ( 0 ) = 0 , τ ′ ( x ) > 0 $\tau (0)=0,~\tau ^{\prime}(x)>0$ and τ ″ ( x ) $\tau^{\prime\prime}(x)$ is bounded for all x ∈ [ 0 , ∞ ) $x\in {}[0,\infty)$ . We study the degree of approximation of these operators by means of the Peetre K-functional and the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja-type theorems are also established in terms of the first order Ditzian-Totik modulus of smoothness.

Keywords

• Baskakov operator
• Ditzian-Totik modulus of smoothness
• rate of convergence
• Voronovskaja-type theorem