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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 \([-\pi T, \pi T]\) for some \(T>0,\) i.e., the (normalized) Fourier transform \(\widehat {f}(v):=(1/\sqrt {2\pi }) \int _{\mathbb {R}} f(u) e^{-ivu}du\) vanishes outside \([-\pi T, \pi T],\) then f can be completely reconstructed for all \(u\in \mathbb {R}\) from its sampled values \(f(k/T)\) taken at the equally spaced nodes \(k/T\) with \(k\in \mathbb {Z}\) .

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Exponential Sampling Theory

  • 章节

  • 发表日期：

    2025年

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  • Carlo Bardaro,
  • Paul L. Butzer,
  • Ilaria Mantellini,
  • Gerhard Schmeisser

摘要

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 \([-\pi T, \pi T]\) for some \(T>0,\) i.e., the (normalized) Fourier transform \(\widehat {f}(v):=(1/\sqrt {2\pi }) \int _{\mathbb {R}} f(u) e^{-ivu}du\) vanishes outside \([-\pi T, \pi T],\) then f can be completely reconstructed for all \(u\in \mathbb {R}\) from its sampled values \(f(k/T)\) taken at the equally spaced nodes \(k/T\) with \(k\in \mathbb {Z}\) .