Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections

DOI10.1007/s13373-014-0057-3zbMath1306.30009OpenAlexW2128424525WikidataQ59394164 ScholiaQ59394164MaRDI QIDQ5172859

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

Publication date: 5 February 2015

Published in: Bulletin of Mathematical Sciences (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s13373-014-0057-3

zbMATH Keywords

sampling theoremEuler-Maclaurin summation formulaRiemann's zeta functionreproducing kernel formulaParseval formulaPoisson's summation formulabandlimited and non-bandlimited functions

Mathematics Subject Classification ID

Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable (30D05) Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type (42A38) Remainders in approximation formulas (41A80) Sampling theory in information and communication theory (94A20) Representations of entire functions of one complex variable by series and integrals (30D10)

Related Items (8)

Quadrature formulae for the positive real axis in the setting of Mellin analysis: sharp error estimates in terms of the Mellin distance ⋮ Sobolev spaces of fractional order, Lipschitz spaces, readapted modulation spaces and their interrelations; applications ⋮ A retrospective on research visits of Paul Butzer's Aachen research group to Eastern Europe and Tenerife ⋮ Pure point diffraction and Poisson summation ⋮ The Butzer-Kozakiewicz article on Riemann derivatives of 1954 and its influence ⋮ Sampling Theory in a Fourier Algebra Setting ⋮ The Mellin–Parseval formula and its interconnections with the exponential sampling theorem of optical physics ⋮ Exponential sampling series: convergence in Mellin-Lebesgue spaces

Cites Work

• Sampling in reproducing kernel Banach spaces
• The summation formulae of Euler-Maclaurin, Abel-Plana, Poisson, and their interconnections with the approximate sampling formula of signal analysis
• Mellin transforms and asymptotics: Harmonic sums
• The Euler-MacLaurin summation formula, the sampling theorem, and approximate integration over the real axis
• On the approximate form of Kluvánek's theorem
• A zero-density theorem for the Riemann zeta-function
• The Riemann zeta-function. Transl. from the Russian by Neal Koblitz
• The sampling theorem, Dirichlet series and Hankel transforms
• Sampling of Paley-Wiener functions on stratified groups
• Another proof of the functional equation for the Riemann zeta function
• A direct approach to the Mellin transform
• The Whittaker-Kotel'nikov-Shannon theorem, spectral translates and Plancherel's formula
• Sampling and reconstruction of signals in a reproducing kernel subspace of \(L^p(\mathbb R^d)\)
• A converse to Kluvánek's theorem
• Existence of Borel transversals in groups
• On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem
• Summation formulas and band-limited signals
• Polynomial splines and eigenvalue approximations on quantum graphs
• A short proof of the functional equation of the Riemann zeta function
• The classical and approximate sampling theorems and their equivalence for entire functions of exponential type
• Easy proofs of Riemann’s functional equation for $\zeta (s)$ and of Lipschitz summation
• Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces
• Convolution sampling and reconstruction of signals in a reproducing kernel subspace
• The sampling theorem, Poisson's summation formula, general Parseval formula, reproducing kernel formula and the Paley–Wiener theorem for bandlimited signals – their interconnections
• Approximating signals in the abstract
• Sur l'équation fonctionnelle de Riemann et la formule sommatoire de Poisson
• Sampling Expansions in Reproducing Kernel Hilbert and Banach Spaces
• Sampling in Paley-Wiener spaces on combinatorial graphs
• The Sampling Theorem, DIRICHLET Series and BESSEL Functions
• A Generalized Sampling Theorem for Locally Compact Abelian Groups
• Analytic Number Theory
• A sampling theorem on homogeneous manifolds
• Paley–Wiener spaces and their reproducing formulae
• Theory of Reproducing Kernels
• A first course in harmonic analysis
• Sampling of band-limited vectors
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