图书简介

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 Kloosters; 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

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