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DOI: 10.23671/VNC.2016.4.5995

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

Umakhanov A. Y. , Sharapudinov I. I.
Vladikavkaz Mathematical Journal 2016. Vol. 18. Issue 4.
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.
Keywords: nonlinear system of integral equations, Hammerstein--Voltera type operator, iteration, monotonisity, primitive matrix, summerable solution
Language: Russian Download the full text  
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