Sampling Theory in Fourier and Signal Analysis: Foundations

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• 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 ; 13.3 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