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On the Duality of Regular and Local Functions

Fischer, Jens (2017) On the Duality of Regular and Local Functions. Mathematics, 5 (41), pp. 1-14. Multidisciplinary Digital Publishing Institute (MDPI). doi: 10.3390/math5030041. ISSN 2227-7390.

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Official URL: http://www.mdpi.com/2227-7390/5/3/41


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 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.

Item URL in elib:https://elib.dlr.de/113670/
Document Type:Article
Title:On the Duality of Regular and Local Functions
AuthorsInstitution or Email of AuthorsAuthor's ORCID iDORCID Put Code
Fischer, JensUNSPECIFIEDhttps://orcid.org/0000-0002-8987-0859UNSPECIFIED
Date:9 August 2017
Journal or Publication Title:Mathematics
Refereed publication:Yes
Open Access:Yes
Gold Open Access:Yes
In ISI Web of Science:Yes
Page Range:pp. 1-14
Publisher:Multidisciplinary Digital Publishing Institute (MDPI)
Keywords:generalized functions, tempered distributions, regular functions, local functions, regularization-localization duality, regularity, Heisenberg's uncertainty principle
HGF - Research field:Aeronautics, Space and Transport
HGF - Program:Space
HGF - Program Themes:Earth Observation
DLR - Research area:Raumfahrt
DLR - Program:R EO - Earth Observation
DLR - Research theme (Project):R - Aircraft SAR
Location: Oberpfaffenhofen
Institutes and Institutions:Microwaves and Radar Institute > SAR Technology
Deposited By: Fischer, Jens
Deposited On:10 Aug 2017 16:36
Last Modified:14 Dec 2019 04:24

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