Abstract
Recently, a mixed hybrid operator, generalizing the well-known Phillips operators and Baskakov–Szász type operators, was introduced. In this paper, we study Bézier variant of these new operators. We investigate the degree of approximation of these operators by means of the Lipschitz class function, the modulus of continuity, and a weighted space. We study a direct approximation theorem by means of the unified Ditzian–Totik modulus of smoothness. Furthermore, the rate of convergence for functions having derivatives of bounded variation is discussed.
Highlights
For a continuous function h on [0, 1], Bernstein [1] defined a linear positive operator in order to provide a very simple and elegant proof of the Weierstrass approximation theorem, namely n Bn(h; x) =n xk(1 – x)n–kh k, k n x ∈ [0, 1]. k=0In order to approximate continuous functions on [0, ∞), Szász [2] introduced the operator Sn(h; x) = ∞k e–nxh k! k n, k=0 (1.1)
Acar et al [22] considered the Bézier variant of Bernstein–Durrmeyer type operators and studied the degree of approximation of functions having derivative of bounded variation
The aim of this paper is to investigate the weighted approximation properties and a direct approximation result by means of the Ditzian–Totik modulus of smoothness ωφτ (h; t), 0 ≤ τ ≤ 1, and the rate of convergence for functions having a derivative of bounded variation for the operators given by (1.4)
Summary
In order to approximate continuous functions on [0, ∞), Szász [2] introduced the operator Acar et al [22] considered the Bézier variant of Bernstein–Durrmeyer type operators and studied the degree of approximation of functions having derivative of bounded variation. The order of approximation of summation-integral type operators for functions with derivatives of bounded variation is estimated in [13, 23,24,25,26,27].
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