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Mathematics 2017, 5(3), 41; https://doi.org/10.3390/math5030041

On the Duality of Regular and Local Functions

German Aerospace Center (DLR), Microwaves and Radar Institute, 82234 Wessling, Germany
Academic Editor: Hari M. Srivastava
Received: 23 May 2017 / Revised: 14 July 2017 / Accepted: 28 July 2017 / Published: 9 August 2017
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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 also inverse of each other. 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. View Full-Text
Keywords: generalized functions; tempered distributions; regular functions; local functions; regularization–localization duality; regularity; Heisenberg’s uncertainty principle generalized functions; tempered distributions; regular functions; local functions; regularization–localization duality; regularity; Heisenberg’s uncertainty principle
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Fischer, J.V. On the Duality of Regular and Local Functions. Mathematics 2017, 5, 41.

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