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Mathematics 2015, 3(2), 299-318; doi:10.3390/math3020299

On the Duality of Discrete and Periodic Functions

Studies conducted at the Institute of Mathematics, Ludwig Maximilians University Munich, 80333 Munich, Germany
Academic Editor: Hari M. Srivastava
Received: 8 October 2014 / Accepted: 22 April 2015 / Published: 30 April 2015
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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. View Full-Text
Keywords: Poisson Summation Formula, discreteness, periodicity, sampling, replicating, Schwartz space, tempered distributions, generalized functions, discrete functions Poisson Summation Formula, discreteness, periodicity, sampling, replicating, Schwartz space, tempered distributions, generalized functions, discrete functions
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Fischer, J.V. On the Duality of Discrete and Periodic Functions. Mathematics 2015, 3, 299-318.

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