On the Duality of Regular and Local Functions

Jens V. Fischer
Additional contact information
Jens V. Fischer: German Aerospace Center (DLR), Microwaves and Radar Institute, 82234 Wessling, Germany

Mathematics, 2017, vol. 5, issue 3, 1-14

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.

Keywords: generalized functions; tempered distributions; regular functions; local functions; regularization–localization duality; regularity; Heisenberg’s uncertainty principle (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2017
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/5/3/41/pdf (application/pdf)
https://www.mdpi.com/2227-7390/5/3/41/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:5:y:2017:i:3:p:41-:d:107674

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().