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The sampling theorem and several equivalent results in analysis. (English) Zbl 1030.41010
Summary: First we show that several fundamental results on functions from the Bernstein spaces $B^p_\sigma$ (such as Bernstein’s inequality and the reproducing formula) can be deduced from a weak form of the classical sampling theorem. In § 3 we discuss the mutual equivalence of the sampling theorem, the derivative sampling theorem and a harmonic function sampling theorem. In §§ 4-6 we discuss connections between the sampling theorem and various important results in complex analysis and Fourier analysis. Our considerations include Cauchy’s integral formula, Poisson’s summation formula, a Gaussian integral, certain properties of weighted Hermite polynomials, Plancherel’s theorem, the maximum modulus principle, and the Phragmén-Lindelöf principle.

41A17Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
94C10Switching theory, application of Boolean algebra; Boolean functions
42A38Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
30D15Special classes of entire functions; growth estimates
30E10Approximation in the complex domain
30E20Integration, integrals of Cauchy type, etc. (one complex variable)
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