Time-Domain Concentration and Approximation of Computable Bandlimited Signals

Abstract

We study the time-domain concentration of bandlimited signals form a computational point of view. To this end we employ the concept of Turing computability that exactly describes what can be theoretically computed on a digital machine. A previous definition of computability for bandlimited signals is based on the idea of effective approximation with finite Shannon sampling series. In this paper we provide a different definition that uses the time-domain concentration of the signals. For computable bandlimited signals with finite L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> -norm, we prove that both definitions are equivalent. We further show that local computability together with the computability of the L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> -norm imply the computability of the signal itself. This provides a simple test for computability.