SOME APPROXIMATION PROPERTIES OF MODIFIED BASKAKOV STANCU OPERATORS

Anshul Srivastava

Abstract: In the present paper, we prove a global direct theorem for the modified Baskakovstancu operators in terms of Ditzian-Totik modulus of smoothness. Here, we have modified our operators by taking weight function of Beta operators and then generalizing it as stancu type generalized operators. We will also see that taking weight function of Beta operators will give better approximation. We study a global direct theorem using simultaneous approximation for ourstancu type generalized operator in ???? 0, 8 . Here, first we estimate recurrence relation for moments and then develop some global direct results by making our stancu type generalized operators positive using differential and integral operators. In this paper, our effort is to give better global approximation for our stancu type generalized operator than the earlier integral modifications of Baskakov operators studied by various authors. Here, we will extend our results for the whole interval 0, 8 . In this paper, we will also make use of the fact that second modulus of smoothness introduced by Ditzian-Totik is equivalent to modified k-functional and ???? ?? [0, 8) is not contained in ??1 ?? [0, 8) for obtaining results. Here, Riesz-Thorin theorem and Leibnitz theorem is used extensively for doing simultaneous approximation. We have also used Fubini’s theorem for obtaining results.

Keywords: Stancu type generalization, simultaneous approximation, modulus of smoothness

DOI: https://doi.org/10.15623/ijret.2015.0408017