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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
\(\bullet\) sampling of band-limited functions; Poisson’s summation formula
\(\bullet\) sampling of derivatives and Hilbert transforms; multichannel sampling
\(\bullet\) error estimates; truncation, amplitude, time-jitter errors
\(\bullet\) sampling of not necessary band-limited functions; aliasing error
\(\bullet\) Kramer’s generalized sampling theorem
\(\bullet\) sampling of special functions and Riemann zeta function
\(\bullet\) generalized sampling series
\(\bullet\) behaviour of sampling series at jump discontinuities
\(\bullet\) sampling in abstract spaces
\({\phantom \bullet}\) – Hilbert space methods
\({\phantom \bullet}\) – sampling on locally compact abelian groups.
For the entire collection see [Zbl 0947.00008].

MSC:

94C10 Switching theory, application of Boolean algebra; Boolean functions (MSC2010)
42A99 Harmonic analysis in one variable
42C99 Nontrigonometric harmonic analysis
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