Article
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On the Duality of Regular and Local Functions
Version 1
: Received: 23 May 2017 / Approved: 24 May 2017 / Online: 24 May 2017 (08:34:46 CEST)
Version 2 : Received: 16 July 2017 / Approved: 17 July 2017 / Online: 17 July 2017 (06:25:18 CEST)
Version 2 : Received: 16 July 2017 / Approved: 17 July 2017 / Online: 17 July 2017 (06:25:18 CEST)
A peer-reviewed article of this Preprint also exists.
Fischer, J.V. On the Duality of Regular and Local Functions. Mathematics 2017, 5, 41. Fischer, J.V. On the Duality of Regular and Local Functions. Mathematics 2017, 5, 41.
Abstract
In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are furthermore the inverses of one another. While Poisson’s summation formula expresses a duality between discretization and periodization, Heisenberg’s uncertainty principle expresses a duality between regularization and localization. We define regularization and localization on generalized functions and show that the Fourier transform of regular functions are local functions and, vice versa, the Fourier transform of local functions are regular functions.
Keywords
generalized functions; tempered distributions; regular functions; local functions; regularization-localization duality; regularity; Heisenberg’s uncertainty principle
Subject
Computer Science and Mathematics, Analysis
Copyright: 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.
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