On the Distributional Fourier Duality and Its Applications

Sampling theorems for bandlimited functions or distributions are obtained by exploiting the topological isomorphism between the space E′(R) of distributions of compact support on R and the Paley–Wiener spacePWof entire functions satisfying an estimate of the form |f(z)|≤A(1+|z|)NeB|Imz|for some cons...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 1998-11, Vol.227 (1), p.43-54
Main Authors: Garcı́a, Antonio G, Moro, Julio, Hernández-Medina, Miguel Ángel
Format: Article
Language:English
Subjects:
distributions of compact support
Exact sciences and technology
Fourier analysis
Fourier duality, Paley–Wiener spaces
Mathematical analysis
Mathematics
Sciences and techniques of general use
Special functions
uniform and nonuniform sampling
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Summary:Sampling theorems for bandlimited functions or distributions are obtained by exploiting the topological isomorphism between the space E′(R) of distributions of compact support on R and the Paley–Wiener spacePWof entire functions satisfying an estimate of the form |f(z)|≤A(1+|z|)NeB|Imz|for some constantsA,B,N≥0. We obtain sampling theorems forfinPWby expanding its Fourier transformTin a series converging in the topology of E′(R) and whose coefficients are samples taken fromf. By Fourier duality, we obtain a sampling theorem forfin the spacePW. These sampling expansions converge, in fact, uniformly on compact sets of C, since convergence in the topology ofPWimplies uniform convergence on compact sets of C. This procedure allows us to recover previous sampling theorems in a unified way. We also present further expansions of Paley–Wiener functions obtained by expanding their Fourier transform as a series involving Legendre or Hermite polynomials.
ISSN:0022-247X
1096-0813
DOI:10.1006/jmaa.1998.6067