Functions with Restricted Transforms

Abstract

The Fourier transforms of functions in signal processing systems typically have support in certain fixed regions within the frequency domain. Such band-limited functions are constrained in ways that are perhaps surprising at first. The sampling theorem provides a method for recovering functions of one variable that are band-limited to an interval from a discrete set of samples. This motivates a more general study of functions whose Fourier transforms have restricted support. Further light is shed on this phenomenon by the fact that such one-variable band-limited functions are entire functions satisfying certain integral and pointwise growth conditions. In fact, the Paley-Wiener theorem actually characterizes functions band-limited to an interval by these conditions. It is another indication that band-limited functions are somewhat rigid in their structure. Also of extreme importance, are the constraints on the number of linearly independent band-limited functions that can be ‘packed’ into a given time interval. A rigorous exploration, due to Landau, Slepian and Pollak, of this time-bandwidth intuition will be given in the last section of this chapter.