On the behavior of Shannon's sampling series for bounded signals with applications

Abstract

In this paper we discuss the interplay between discrete-time and continuous-time signals and the question, whether certain properties of the signal in one domain carry over to the other domain. The Shannon sampling series and the more general Valiron interpolation series are the appropriate means to obtain the continuous-time, bandlimited signal out of its samples, i.e., the discrete-time signal. We study the effects of the projection operator, that projects the discrete-time signal on the positive time axis and, closely related, the differences in the convergence behavior of the symmetric and asymmetric Shannon sampling series. It is well known, that the space of discrete-time signals with finite energy and the space of continuous-time, bandlimited signals with finite energy are isomorphic. Thus, discrete-time and continuous-time, bandlimited signals with finite energy can be used interchangeably. This interchangeability is not restricted to finite energy signals. It is valid for a considerably larger class, but, as we show by stating an explicit example, not for the space of bounded signals: Even if the discrete-time signal is bounded, the corresponding bandlimited interpolation can be unbounded. In addition to the fact that the Shannon sampling series diverges for this signal, we prove that for this signal there is no bounded, bandlimited interpolation at all. Furthermore, by using the Banach–Steinhaus theorem we show that in a certain sense “almost all” signals have this property. Finally, some implications for practical applications are discussed.