Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

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)

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.


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.


generalized functions; tempered distributions; regular functions; local functions; regularization-localization duality; regularity; Heisenberg’s uncertainty principle


Computer Science and Mathematics, Analysis

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