On the Duality of Discrete and Periodic Functions

Jens V. Fischer
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Jens V. Fischer: Studies conducted at the Institute of Mathematics, Ludwig Maximilians University Munich, 80333 Munich, Germany

Mathematics, 2015, vol. 3, issue 2, 1-20

Abstract: Although versions of Poisson’s Summation Formula (PSF) have already been studied extensively, there seems to be no theorem that relates discretization to periodization and periodization to discretization in a simple manner. In this study, we show that two complementary formulas, both closely related to the classical Poisson Summation Formula, are needed to form a reciprocal Discretization-Periodization Theorem on generalized functions. We define discretization and periodization on generalized functions and show that the Fourier transform of periodic functions are discrete functions and, vice versa, the Fourier transform of discrete functions are periodic functions.

Keywords: Poisson Summation Formula; discreteness; periodicity; sampling; replicating; Schwartz space; tempered distributions; generalized functions; discrete functions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2015
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