Cardinal sine series, oversampling, and periodic distributions

Abstract:

Suppose $u(n/\rho )$, $n=0, \pm 1, \pm 2, \ldots$, are samples of a frequency band limited function $u(z)$ and $\rho$ is greater than the Nyquist rate. If the even part of these samples, $u_e(n/\rho )$, has less than quadratic growth and the odd part, $u_o(n/\rho )$, has less than linear growth as $n \to \pm \infty$, then we show that the corresponding cardinal sine series, that plays an essential role in the Whittaker-Kotelnikov-Shannon sampling theorem, converges uniformly to $u(z)$ on compact subsets of the complex plane. An appropriately adapted version of the technique used to prove this allows us to obtain a result concerning the local convergence of the Fourier series of periodic distributions.