Abstract
We give a Bézier variant of Baskakov-Durrmeyer-type hybrid operators in the present article. First, we obtain the rate of convergence by using Ditzian-Totik modulus of smoothness and also for a class of Lipschitz function. Then, weighted modulus of continuity is investigated too. We study the rate of point-wise convergence for the functions having a derivative of bounded variation. Furthermore, we establish the quantitative Voronovskaja-type formula in terms of Ditzian-Totik modulus of smoothness at the end.
Highlights
Many approximating operators have been introduced under certain conditions and with different parameters too
Among interesting studies realized in this context, we cite those based on the Baskakov-Kantorovitch-type operators in the generalized form defined as, for f ∈ L1ð1⁄20, 1Þ
Since (1) and (2) are evident, we prove only the assertion (3)
Summary
Many approximating operators have been introduced under certain conditions and with different parameters too. Many researchers have later generalized and modified these introduced operators and discussed various approximating properties of these operators. In 1957, Baskakov [1] introduced and studied such a class of positive linear operators, called Baskakov operators defined on the positive semiaxis. Among interesting studies realized in this context, we cite those based on the Baskakov-Kantorovitch-type operators in the generalized form (the original operator given by Kantorovich in [3]) defined as, for f ∈ L1ð1⁄20, 1Þ (the class of Lebesgue integrable functions on 1⁄20, 1), BKnð f ; xÞ = 〠. The miscellaneous Bézier variant of operators is crucial subject matter in approximation theory. We will be mainly interested to the Bézier variant operator type based on those of Baskakov-Durrmeyer defined as follows: G v,θ n,ρ ð f.
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