Interpolation of Functions by the Whittaker Sums and their Modifications: Conditions for Uniform Convergence
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.