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Zusatztext ...the text is written by use of LATEX and its beautiful graphics reveal the power and the advantages of this system. Klappentext 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. Zusammenfassung This monograph explores 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 and their bases, it covers a wide variety of topics. Inhaltsverzeichnis 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 13.3: 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-di...
