Sampling Theory in Fourier and Signal Analysis: Foundations

Containing important new material unavailable previously in book form, this book covers a wide variety of topics which will be great interest to applied mathematicians and engineers. Introducing the main ideas, background material is provided on Fourier analysis, Hilbert spaces, and their bases, before the book moves on to discuss more complex topics and their applications.

Table of Contents:
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 Kloösters 6: Bernstein and Paley-Weiner spaces 6.1: Convolution and the cardinal series 6.2: Sampling and entire functions of polynomial growth 6.3: Paley-Weiner spaces 6.4: The cardinal series for Paley-Weiner spaces 6.5: The space ReH1 6.6: The ordinary Paley-Weiner space and its reproducing kernel 6.7: A convergence principle for general Paley-Weiner spaces 7: More about Paley-Weiner spaces 7.1: Paley-Weiner theorems - a review 7.2: Bases for Paley-Weiner spaces 7.3: Operators on the Paley-Weiner space 7.4: Oscillatory properties of Paley-Weiner functions 8: Kramer's lemma 8.1: Kramer's Lemma 8.2: The Walsh sampling therem 9: Contour integral methods 9.1: The Paley-Weiner theorem 9.2: Some formulae of analysis and their equivalence 9.3: A general sampling theorem 10: Ireggular sampling 10.1: Sets of stable sampling, of interpolation and of uniqueness 10.2: Irregular sampling at minimal rate 10.3: Frames and over-sampling 11: Errors and aliasing 11.1: Errors 11.2: The time jitter error 11.3: The aliasing error 12: Multi-channel sampling 12.1: Single channel sampling 12.3: Two channels 13: Multi-band sampling 13.1: Regular sampling 13.2 Optimal regular sampling: 13.3: An algorithm for the optimal regular sampling rate 13.4: Selectively tiled band regions 13.5: Harmonic signals 13.6: Band-ass sampling 14: Multi-dimensional sampling 14.1: Remarks on multi-dimensional Fourier analysis 14.2: The rectangular case 14.3: Regular multi-dimensional sampling 15: Sampling and eigenvalue problems 15.1: Preliminary facts 15.2: Direct and inverse Sturm-Liouville problems 15.3: Further types of eigenvalue problem - some examples 16: Campbell's generalised sampling theorem 16.1: L.L. Campbell's generalisation of the sampling theorem 16.2: Band-limited functions 16.3: Non band-limited functions - an example 17: Modelling, uncertainty and stable sampling 17.1: Remarks on signal modelling 17.2: Energy concentration 17.3: Prolate Spheroidal Wave functions 17.4: The uncertainty principle of signal theory 17.5: The Nyquist-Landau minimal sampling rate

Review :
This book is an excellent introduction into various aspects of sampling theory, written by a long-standing expert in this field. The book covers most of the standard topics in sampling theory....The book is aimed towards a mathematical audience and may form the basis for a one semester graduate corse. But it is also highly recommended for those of the scientific community, who wish to inform themselves about the basics of the theory, particularly those in mathematics, applied mathematics, and engineering.