Abstract
First we show that several fundamental results on functions from theBernstein spaces \(B_\sigma ^p \) (such as Bernstein's inequality andthe reproducing formula) can be deduced from a weak form of the classicalsampling theorem. In §3 we discuss the mutual equivalence of thesampling theorem, the derivative sampling theorem and a harmonic functionsampling theorem. In §§4–6 we discuss connections between thesampling theorem and various important results in complex analysis andFourier analysis. Our considerations include Cauchy's integral formula,Poisson's summation formula, a Gaussian integral, certain properties ofweighted Hermite polynomials, Plancherel's theorem, the maximum modulusprinciple, and the Phragmén–Lindelöf principle.
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Higgins, J.R., Schmeisser, G. & Voss, J.J. The Sampling Theorem and Several Equivalent Results in Analysis. Journal of Computational Analysis and Applications 2, 333–371 (2000). https://doi.org/10.1023/A:1010164717587
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DOI: https://doi.org/10.1023/A:1010164717587