Preprint Article Version 1 This version not peer reviewed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.

Journal reference: Mathematics 2017, 5, 41
DOI: 10.3390/math5030041

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.

Subject Areas

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

Readers' Comments and Ratings (1)

Comment 1
Received: 30 June 2017
Commenter: Jens Fischer
The commenter has declared there is no conflict of interests.
Comment: If you read this paper, please send me your suggestions or questions to jens.fischer@ieee.org, anonymously, if you like. It would help me a lot to learn what is needed to be better understood. Thank you very much. Best regards, JF.
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