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On the Reversibility of Discretization

Fischer, Jens and Stens, Rudolf (2020) On the Reversibility of Discretization. Mathematics, 8 (619), pp. 1-21. Multidisciplinary Digital Publishing Institute (MDPI). DOI: 10.3390/math8040619 ISSN 2227-7390

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Official URL: https://www.mdpi.com/2227-7390/8/4/619

Abstract

Discretization usually denotes the operation of mapping continuous functions to infinite or finite sequences of discrete values. It may also mean to map the operation itself from one that operates on functions to one that operates on infinite or finite sequences. Advantageously, these two meanings coincide within the theory of generalized functions. Discretization moreover reduces to a simple multiplication. It is known, however, that multiplications may fail. In our previous studies, we determined conditions such that multiplications hold in the tempered distributions sense and, hence, corresponding discretizations exist. In this study, we determine, vice versa, conditions such that discretizations can be reversed, i.e., functions can be fully restored from their samples. The classical Whittaker-Kotelnikov-Shannon (WKS) sampling theorem is just one particular case in one of four interwoven symbolic calculation rules deduced below.

Item URL in elib:https://elib.dlr.de/134665/
Document Type:Article
Title:On the Reversibility of Discretization
Authors:
AuthorsInstitution or Email of AuthorsAuthor's ORCID iD
Fischer, JensJens.Fischer (at) dlr.dehttps://orcid.org/0000-0002-8987-0859
Stens, Rudolfstens (at) matha.rwth-aachen.deUNSPECIFIED
Date:17 April 2020
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:8
DOI :10.3390/math8040619
Page Range:pp. 1-21
Publisher:Multidisciplinary Digital Publishing Institute (MDPI)
ISSN:2227-7390
Status:Published
Keywords:regularization; localization; truncation; cutoff; finitization; entirization; cyclic dualities; multiplication of distributions; square of the Dirac delta; Whittaker-Kotelnikov-Shannon sampling theorem
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:20 Apr 2020 06:33
Last Modified:15 May 2020 14:15

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