Sampling Theory in Fourier and Signal Analysis: Foundations Volume 1: Foundations

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.

Containing important new material unavailable previously in book form, this book covers a wide variety of topics which will be great interest to applied mathematicians and engineers. Introducing the main ideas, background material is provided on Fourier analysis, Hilbert spaces, and their bases, before the book moves on to discuss more complex topics and their applications.

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-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