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Sampling theory of signal analysis. (English) Zbl 0961.94009
Pier, Jean-Paul (ed.), Development of mathematics 1950-2000. Basel: Birkhäuser. 193-234 (2000).

The paper gives a good short overview of results in sampling theory from the Whittaker-Kotel’nikov-Shannon sampling theorem to recent developments. In particular the authors consider

sampling of band-limited functions; Poisson’s summation formula

sampling of derivatives and Hilbert transforms; multichannel sampling

error estimates; truncation, amplitude, time-jitter errors

sampling of not necessary band-limited functions; aliasing error

Kramer’s generalized sampling theorem

sampling of special functions and Riemann zeta function

generalized sampling series

behaviour of sampling series at jump discontinuities

sampling in abstract spaces

– Hilbert space methods

– sampling on locally compact abelian groups.

MSC:
94C10Switching theory, application of Boolean algebra; Boolean functions
42A99Fourier analysis in one variable
42C99Non-trigonometric Fourier analysis