Sampling Theory in Fourier and Signal Analysis

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