Simultaneous Approximation for Generalized Baskakov-Durrmeyer-Type Operators (original) (raw)

Abstract

The present paper deals with the study of a Durrmeyer-type integral modification of certain modified Baskakov operators. Here we study simultaneous approximation properties for these operators by using the iterative combinations. We obtain an asymptotic formula and an error estimation in terms of higher order modulus of continuity for these operators.

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What are the main properties of the new Baskakov-Durrmeyer operators introduced?add

The newly defined Baskakov-Durrmeyer operators reproduce both constants and linear functions, unlike traditional operators.

How does the order of approximation for these operators compare to existing methods?add

The paper demonstrates that the order of approximation is O(n^{-1}), consistent even for smooth functions.

What is the role of iterative combinations in improving approximation accuracy?add

Iterative combinations, as proposed by Micchelli, enhance the order of approximation beyond standard limits.

What asymptotic behaviors or error estimations are derived in the research?add

The study establishes Voronovskaja-type formulas and provides error estimations for the operators' iterative combinations.

How does this work extend previous studies on generalized Baskakov-type operators?add

This work builds on Wang Li's research, addressing operators that reproduce linear functions, previously unconsidered.

Figures (5)

Proof of Theorem 1.1. By applying Lemma 2.9, we get

Proof of Theorem 1.1. By applying Lemma 2.9, we get

Proof of Theorem 1.2. By Taylor expansion of f, we have

Proof of Theorem 1.2. By Taylor expansion of f, we have

The second term in the right hand side of above expression multiplied by n* tends to zero as n — +00. Since e(t,7) > 0 as t > a, for a given ¢ > 0 there exists a 0 > 0, such that |e(t,x)| < ¢ whenever 0 < |t — 2| < 6. Also, for |é — 2| > 4, we have |e(t,x)(t — «)?**"| < M(1+t)7 for some M > 0, since we can consider

The second term in the right hand side of above expression multiplied by n* tends to zero as n — +00. Since e(t,7) > 0 as t > a, for a given ¢ > 0 there exists a 0 > 0, such that |e(t,x)| < ¢ whenever 0 < |t — 2| < 6. Also, for |é — 2| > 4, we have |e(t,x)(t — «)?**"| < M(1+t)7 for some M > 0, since we can consider

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References (10)

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