The sampling theorem, Poisson's summation formula, general Parseval formula, reproducing kernel formula and the Paley–Wiener theorem for bandlimited signals – their interconnections

DOI10.1080/00036811003627567zbMath1217.30025OpenAlexW2125021786WikidataQ58272027 ScholiaQ58272027MaRDI QIDQ2995522

J. R. Higgins, Rudolf L. Stens, Gerhard Schmeisser, Paul L. Butzer, Paulo Jorge S. G. Ferreira

Publication date: 21 April 2011

Published in: Applicable Analysis (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1080/00036811003627567

zbMATH Keywords

Paley-Wiener theorembandlimited signalsPoisson summation formulaValiron-Tschakaloff sampling/interpolation formulaWhittaker-Kotel'nikov-Shannon sampling theorem

Mathematics Subject Classification ID

General harmonic expansions, frames (42C15) Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) (46E22) Sampling theory in information and communication theory (94A20) Representations of entire functions of one complex variable by series and integrals (30D10)

Related Items (15)

Whittaker-Kotel'nikov-Shannon approximation of \(\phi\)-sub-Gaussian random processes ⋮ The classical and approximate sampling theorems and their equivalence for entire functions of exponential type ⋮ Sobolev spaces of fractional order, Lipschitz spaces, readapted modulation spaces and their interrelations; applications ⋮ Sampling and interpolation for the discrete Hilbert and Kak–Hilbert transforms ⋮ Shift-invariant and sampling spaces associated with the special affine Fourier transform ⋮ Basic relations valid for the Bernstein space \(B^{p}_{\sigma}\) and their extensions to functions from larger spaces with error estimates in terms of their distances from \(B^{p}_{\sigma}\) ⋮ Band-limited scaling and wavelet expansions ⋮ Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections ⋮ The summation formulae of Euler-Maclaurin, Abel-Plana, Poisson, and their interconnections with the approximate sampling formula of signal analysis ⋮ The Mellin–Parseval formula and its interconnections with the exponential sampling theorem of optical physics ⋮ Thoughts on Numerical and Conceptual Harmonic Analysis ⋮ Exponential sampling series: convergence in Mellin-Lebesgue spaces ⋮ Aspects of prediction ⋮ Unnamed Item ⋮ Multivariate sampling-type approximation

Cites Work

• Unnamed Item
• On the approximate form of Kluvánek's theorem
• Summation formulas and band-limited signals
• Classical and approximate sampling theorems; studies in the \(L^{p}(\mathbb R)\) and the uniform norm
• Five short stories about the cardinal series
• Hilbert's Twenty-Fourth Problem
• Mathematical problems
• Elements of Information Theory

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