The sampling theorem, Dirichlet series and Hankel transforms

Abstract

Some very surprising relations between fundamental theorems and formulas of signal analysis, of analytic number theory and of applied analysis are presented. It is shown that generalized forms of the classical Whittaker—Kotelnikov—Shannon sampling theorem as well as of the Brown—Butzer—Splettstößer approximate sampling expansion for non-band-limited signal functions can be deduced via the theory of Dirichlet series with functional equations from a new summation formula for Hankel transforms. This counterpart to Poisson's summation formula is shown to be essentially “equivalent” to the famous functional equation of Riemann's zeta-function, to the “modular relation” of the theta-function, to the Nielsen—Doetsch summation formula for Bessel functions and to the partial fraction expansion of the periodic Hilbert kernel.