Sampling Theory in Fourier and Signal Analysis: Foundations

This book is about the equivalence of signal functions with their sets of values taken at discrete points. Beginning with an introduction to the main ideas, and background material on Fourier analysis and Hilbert spaces amd their bases, the book covers a wide variety of topics including: sampling of Bernstein and Paley-Wiener spaces; Kramer's Lemma and its application to eigenvalue problems; contour integral methods including a proof of the equivalence of the sampling theory; the Poisson summation formula and Cauchy's integral formula; optimal regular, irregular, multi-channel, multi-band and multi dimensional sampling, and Campbell's generalized sampling theorem.