Interpolation of functions by the Whittaker sums and their modifications: conditions for uniform convergence. (Russian. English summary) Zbl 1474.41007

Summary: 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.


41A05 Interpolation in approximation theory
41A45 Approximation by arbitrary linear expressions
42A10 Trigonometric approximation
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
Full Text: MNR


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