Recovery of digitized signals using Slepian functions

The continuous prolate spheroidal wave functions (Slepian functions) were found to be useful for analog signal processing several decades ago. But the digital revolution left them in the dust since they did not seem naturally adapted to discrete analysis. Yet they have many desirable, even unique, properties that originally made them fascinating and could lead to some applications in digital and analog-digital signal processing practice. The simplest such applications involve the conversion of a digital signal to an analog signal and recovery of a bandlimited signal from their values at countable many distinct points on the real line. The Shannon sampling theorem, which is based on the sinc expansions, plays the most important theoretical foundation in such approaches. In this work, by using the natural connection between the Slepian functions and sinc function, several new formulae based on integer values of Slepian functions are developed. These formulae are then used to replace the sinc function in sampling theorems for digitizing bandlimited signals. Finally, they are used to construct analysis and synthesis filter banks for sampled values of a bandlimited signal.