Approximation of Functions by Modified Phillips q − $$ q- $$ Bernstein Operators of Order Two

IF 2 3区数学 Q1 MATHEMATICS, APPLIED

Mathematical Methods in the Applied Sciences Pub Date : 2026-09-25 Epub Date: 2026-07-23 DOI:10.1002/mma.70907

Nilesh Zapadiya, Shailesh Kumar Srivastava

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引用次数: 0

Abstract

In this paper, we introduce modified Phillips $q -$ Bernstein operators of order two, which provide better uniform approximation than Phillips $q -$ Bernstein operators and Phillips $q -$ Durrmeyer Bernstein operators. In this study, we consider functions belonging to the class $C ([0, 1])$ and show that the modified Phillips $q -$ Bernstein operators converge uniformly to all functions in this class using a strengthened version of the Korovkin theorem. Also, we prove a Voronovskaya-type theorem for modified Phillips $q -$ Bernstein operators, which show better approximation than the Phillips $q -$ Bernstein operators. Using the modulus of continuity, we show that the order of approximation of modified Phillips $q -$ Bernstein operators is better than that of Phillips $q -$ Bernstein operators and Phillips $q -$ Durrmeyer Bernstein operators. Finally, we compare the convergence rate of Phillips $q -$ Bernstein operators, modified Phillips $q -$ Bernstein operators, and Phillips $q -$ Durrmeyer Bernstein operators by considering the functions $Ψ (x) = e^{\cos (4 x^{2}) + \sin (x)}$ and $Ψ (x) = 1 - \cos (4 e^{x})$ belonging to $C [0, 1])$ .

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修正Phillips q−$$ q- $$二阶Bernstein算子的函数逼近

本文引入了二阶修正的Phillips q−$$ q- $$ Bernstein算子，它们提供了比Phillips q−$$ q- $$ Bernstein算子和Phillips q−$$ q- $$ Durrmeyer Bernstein算子更好的均匀逼近。在本研究中，我们考虑C ([0，1]) $$ C\left(\left[0,1\right]\right) $$，并利用强化版的Korovkin定理证明了修正的Phillips q−$$ q- $$ Bernstein算子一致收敛于该类中的所有函数。此外，我们还证明了修正的Phillips q−$$ q- $$ Bernstein算子的voronovskaya型定理，它比Phillips q−$$ q- $$ Bernstein算子具有更好的逼近性。利用连续模，我们证明了修正的Phillips q−$$ q- $$ Bernstein算子的近似阶数优于Phillips q−$$ q- $$ Bernstein算子和Phillips q−$$ q- $$ Durrmeyer Bernstein运营商。最后，我们比较了Phillips q−$$ q- $$ Bernstein算子、修正Phillips q−$$ q- $$ Bernstein算子、和Phillips q−$$ q- $$ Durrmeyer Bernstein算子，考虑Ψ (x) = e cos (4 × 2) + sin (x) $$ \Psi (x)={e}^{\cos \left(4{x}^2\right)+\sin (x)} $$和Ψ (x) = 1−cos4ex$$ \Psi (x)=1-\cos \left(4{e}^x\right) $$属于C [0,1]) $$ C\left[0,1\right]\Big) $$。

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来源期刊

Mathematical Methods in the Applied Sciences 数学-应用数学

CiteScore

4.90

自引率

6.90%

发文量

798

审稿时长

6 months

期刊介绍： Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.