Cardinal sine series: Oversampling and non-existence

Abstract

Growth conditions are given on the samples f(n), n = 0, ±1, ±2, ..., of an entire function f(z) of exponential type less than π that imply that the corresponding cardinal sine series converges. These conditions are the least restrictive of their kind that are possible. Furthermore, an example is provided of an entire function f(z) of exponential type π that is bounded on the real axis and whose corresponding cardinal sine series fails to converge.