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 Yet they have many desirable, even unique, properties that originally made them fascinating and could lead to some applications in digital and analog-digital signal processing practice.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.

It is well known that the Shannon sampling theorem can be put into a wavelet context. But it has also been shown that for most wavelets, a non-negative summability function for the associated subspaces exists. Translation of these are used in a series with sampling coefficients to get a "hybrid" series each of whose terms is non-negative. Such series converge, avoid integration, and eliminate the Gibbs phenomenon.

It is well known that the Shannon (1949) sampling theorem can be put into a wavelet context. But it has also been shown that for most wavelets, a sampling theorem for the associated subspaces exists. There is even a non-uniform sampling theorem as in the Shannon case. In general the bounds on the deviations from uniform are not as easy to specify in this case. No simple Kadec 1/4 theorem holds except in special cases (such as the Franklin case where the bound is 1/2). For a particular class, the Meyer (1990) wavelets, which are bandlimited but with a smooth spectrum, a similar bound is sometimes obtainable. Unfortunately, it is much smaller that 1/4.

The classical Shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory. Under weak hypotheses, these subspaces are first shown to have a Riesz basis formed from the reproducing kernels. These in turn are used to construct the sampling sequences. Examples are given.>