Behavior of Shannon's Sampling Series 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. Furthermore, we investigate the symmetric sampling series and the behavior of the non-symmetric sampling series, which follows from the properties of the projection operator. 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 not for the space of bounded signals: Even if the discrete-time signal is bounded, the corresponding bandlimited interpolation can be unbounded. For the proof we explicitly state such a bounded discrete-time signal. Furthermore, we show not only that the Shannon sampling series diverges for this signal, but also that there is no bounded, bandlimited interpolation at all.