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.
Highlights
Versions of Poisson’s summation formula have already been the subject of many publications and it is clear that it plays a very central role in mathematics, physics and electrical engineering.Many publications describe the interaction of discretization and periodization
In some publications we find a clear statement that the Poisson’s Summation Formula (PSF) is linked to both discretization and periodization [1,2,3], and in rare cases we find that the PSF is a discretization-periodization theorem [4]
We Mathematics 2015, 3 show that two complementary formulas, very closely related to Poisson’s classical summation formula, are needed to form a complete, i.e., reciprocal, discretization-periodization theorem in the tempered distribution sense
Summary
Versions of Poisson’s summation formula have already been the subject of many publications and it is clear that it plays a very central role in mathematics, physics and electrical engineering. We Mathematics 2015, 3 show that two complementary formulas, very closely related to Poisson’s classical summation formula, are needed to form a complete, i.e., reciprocal, discretization-periodization theorem in the tempered distribution sense. Benedetto and Zimmermann investigated the validity of Poisson’s summation formula in several classical spaces in analysis including the space of tempered distributions [3,16]. They define sampling and periodization operators and, amongst others, characterize uniform sampling as the Fourier transform of periodization. We define discretization and periodization as operations on suitable subspaces of tempered distributions and in Section 5 we hope to motivate the reader to apply them to already known results in textbooks.
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