Description
Book SynopsisThis 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.
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Table of Contents1. 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 ; 13.3 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
