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Vladikavkaz. Mat. Zh., 2016, Volume 18, Number 4, Pages 61–70 (Mi vmj598)  

This article is cited in 4 scientific papers (total in 4 papers)

Interpolation of functions by the Whittaker sums and their modifications: conditions for uniform convergence

A. Y. Umakhanovab, I. I. Sharapudinovbc

a Daghestan Scientific Centre of Russian Academy of Sciences, Makhachkala
b Daghestan State Pedagogical University
c Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences

Abstract: We consider truncated Whittaker–Kotel'nikov–Shannon operators also known as sinc-operators. Conditions on continuous functions $f$ that guarantee uniform convergence of sinc-operators to such functions are obtained. It is shown that if a function is absolutely continuous, satisfies Dini–Lipschitz condition and vanishes at the end of the segment $[0,\pi]$, then sinc-operators converge uniformly to this function. In the case when $f(0)$ or $f(\pi)$ is not zero, sinc-operators lose the property of uniform convergence. For example, it is well known that sinc-operators have no uniform convergence to function identically equal to 1. In connection with this we introduce modified sinc-operators that possess a uniform convergence property for arbitrary absolutely continuous function, satisfying Dini–Lipschitz condition.

Key words: nonlinear system of integral equations, Hammerstein–Voltera type operator, iteration, monotonisity, primitive matrix, summerable solution.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00486_а


Full text: PDF file (247 kB)
References: PDF file   HTML file
UDC: 517.538
Received: 03.03.2016

Citation: A. Y. Umakhanov, I. I. Sharapudinov, “Interpolation of functions by the Whittaker sums and their modifications: conditions for uniform convergence”, Vladikavkaz. Mat. Zh., 18:4 (2016), 61–70

Citation in format AMSBIB
\Bibitem{UmaSha16}
\by A.~Y.~Umakhanov, I.~I.~Sharapudinov
\paper Interpolation of functions by the Whittaker sums and their modifications: conditions for uniform convergence
\jour Vladikavkaz. Mat. Zh.
\yr 2016
\vol 18
\issue 4
\pages 61--70
\mathnet{http://mi.mathnet.ru/vmj598}


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    This publication is cited in the following articles:
    1. A. Yu. Trynin, “A criterion of convergence of Lagrange–Sturm–Liouville processes in terms of one-sided modulus of variation”, Russian Math. (Iz. VUZ), 62:8 (2018), 51–63  mathnet  crossref  isi
    2. A. Yu. Trynin, “Uniform convergence of Lagrange–Sturm–Liouville processes on one functional class”, Ufa Math. J., 10:2 (2018), 93–108  mathnet  crossref  isi
    3. A. Yu. Trynin, “Skhodimost protsessov Lagranzha–Shturma–Liuvillya dlya nepreryvnykh funktsii ogranichennoi variatsii”, Vladikavk. matem. zhurn., 20:4 (2018), 76–91  mathnet  crossref
    4. A. Yu. Trynin, “Sufficient condition for convergence of Lagrange–Sturm–Liouville processes in terms of one-sided modulus of continuity”, Comput. Math. Math. Phys., 58:11 (2018), 1716–1727  mathnet  crossref  crossref  isi
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