Sampling Theory in Fourier and Signal Analysis Volume 1 Foundations [Hardcover]

With much material not previously found in book form, this book fills a gap by discussing the equivalence of signal functions with their sets of values taken at discreet points comprehensively and on a firm mathematical ground. The wide variety of topics begins with an introduction to the main ideas and background material on Fourier analysis and Hilbert spaces and their bases. Other chapters discuss 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. Mathematicians, physicists, and communications engineers will welcome the scope of information found here.

1. An introduction to sampling theory
1.1. General introduction
1.2. Introduction - continued
1.3. The seventeenth to the mid twentieth century - a brief review
1.4. Interpolation and sampling from the seventeenth century to the mid twentieth century - a brief review
1.5. Introduction - concluding remarks
2. Background in Fourier analysis
2.1. The Fourier Series
2.2. The Fourier transform
2.3. Poisson's summation formula
2.4. Tempered distributions - some basic facts
3. Hilbert spaces, bases and frames
3.1. Bases for Banach and Hilbert spaces
3.2. Riesz bases and unconditional bases
3.3. Frames
3.4. Reproducing kernel Hilbert spaces
3.5. Direct sums of Hilbert spaces
3.6. Sampling and reproducing kernels
4. Finite sampling
4.1. A general setting for finite sampling
4.2. Sampling on the sphere
5. From finite to infinite sampling series
5.1. The change to infinite sampling series
5.2. The Theorem of Hinsen and Klo??sters
6. Bernstein and Paley-Weiner spaces
6.1. Convolution and the cardinal series
6.2. SlC