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

A. Y. Umakhanov ; I. I. Sharapudinov

Vladikavkazskij matematičeskij žurnal, Tome 18 (2016) no. 4, pp. 61-70

Résumé

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.