Exponential Sampling Theory

Carlo Bardaro, Paul L. Butzer, Ilaria Mantellini and Gerhard Schmeisser
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Carlo Bardaro: University of Perugia, Department of Mathematics and Informatics
Paul L. Butzer: RWTH Aachen University, Lehrstuhl A für Mathematik
Ilaria Mantellini: University of Perugia, Department of Mathematics and Informatics
Gerhard Schmeisser: Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Department Mathematik

Chapter Chapter 12 in Mellin Analysis, Transform Theory, and Applications, 2025, pp 319-379 from Springer

Abstract: Abstract One of the basic results of signal and Fourier analysis is the Whittaker–Shannon–Kotel’nikov sampling theorem.Whittaker–Kotel’nikov–Shannon (WKS)sampling theoremWKS Whittaker–Kotel’nikov–Shannon (WKS) It states that if f is (Fourier) bandlimited to an interval [ − πT , πT ] $$[-\pi T, \pi T]$$ for some T > 0 , $$T>0,$$ i.e., the (normalized) Fourier transform f ^ ( v ) : = ( 1 ∕ 2 π ) ∫ ℝ f ( u ) e − ivu du $$\widehat {f}(v):=(1/\sqrt {2\pi }) \int _{\mathbb {R}} f(u) e^{-ivu}du$$ vanishes outside [ − πT , πT ] , $$[-\pi T, \pi T],$$ then f can be completely reconstructed for all u ∈ ℝ $$u\in \mathbb {R}$$ from its sampled values f ( k ∕ T ) $$f(k/T)$$ taken at the equally spaced nodes k ∕ T $$k/T$$ with k ∈ ℤ $$k\in \mathbb {Z}$$ .

Date: 2025
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DOI: 10.1007/978-3-031-96672-9_12

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