On a Non-Fourier Generalization of Shannon Sampling Theory

Shannon sampling theory, which is in ubiquitous use in communication engineering and signal processing, shows that in order to record all information contained in a Ω-bandlimited continuous signal it
suffices
to record samples of the signal's amplitude at discrete points in time, namely at the so-called Nyquist rate of 2Ω samples per second. It is clear, however, that for realistic signals is should not be
necessary
to take the samples at this rate throughout. Intuitively, parts of a realistic signal may possess a lower bandwidth-implying that if we do take the samples at the rate 2Ω throughout then many of the sample values will be highly correlated, i.e., wastefully redundant. The challenge to generalize Shannon sampling to allow varying Nyquist rates has therefore become the subject of both research papers and patents. Here we present new results on an approach that makes use of powerful functional analytic methods.