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On Inverses of the Dirac Comb

Fischer, Jens and Stens, Rudolf (2019) On Inverses of the Dirac Comb. Mathematics, 7 (1196), pp. 1-14. Multidisciplinary Digital Publishing Institute (MDPI). ISSN 2227-7390

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Official URL: https://www.mdpi.com/2227-7390/7/12/1196

Abstract

We determine tempered distributions which convolved with a Dirac comb yield unity and tempered distributions which multiplied with a Dirac comb yield a Dirac delta. Solutions of these equations have numerous applications. They allow the reversal of discretizations and periodizations applied to tempered distributions. One of the difficulties is the fact that Dirac combs cannot be multiplied or convolved with arbitrary functions or distributions. We use a theorem of Laurent Schwartz to overcome this difficulty and variants of Lighthills unitary functions to solve these equations. The theorem we prove states that double-sided (time/frequency) smooth partitions of unity are required to neutralize discretizations and periodizations on tempered distributions.

Item URL in elib:https://elib.dlr.de/131858/
Document Type:Article
Title:On Inverses of the Dirac Comb
Authors:
AuthorsInstitution or Email of AuthorsAuthors ORCID iD
Fischer, JensJens.Fischer (at) dlr.dehttps://orcid.org/0000-0002-8987-0859
Stens, Rudolfstens (at) matha.rwth-aachen.deUNSPECIFIED
Date:6 December 2019
Journal or Publication Title:Mathematics
Refereed publication:Yes
Open Access:Yes
Gold Open Access:Yes
In SCOPUS:Yes
In ISI Web of Science:Yes
Volume:7
Page Range:pp. 1-14
Publisher:Multidisciplinary Digital Publishing Institute (MDPI)
ISSN:2227-7390
Status:Published
Keywords:Tempered distribution; partition of unity; unitary function; Poisson Summation Formula; Heisenberg uncertainty principle; Paley-Wiener function; Whittaker-Kotelnikov-Shannon (WKS); arbitrary products; square of the Dirac delta
HGF - Research field:Aeronautics, Space and Transport
HGF - Program:Space
HGF - Program Themes:Earth Observation
DLR - Research area:Raumfahrt
DLR - Program:R EO - Erdbeobachtung
DLR - Research theme (Project):R - Vorhaben Flugzeug-SAR
Location: Oberpfaffenhofen
Institutes and Institutions:Microwaves and Radar Institute > SAR Technology
Deposited By: Fischer, Jens
Deposited On:06 Dec 2019 11:29
Last Modified:14 Dec 2019 04:24

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