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Four Particular Cases of the Fourier Transform

Fischer, Jens (2018) Four Particular Cases of the Fourier Transform. Mathematics, 6 (335), pp. 1-19. Multidisciplinary Digital Publishing Institute (MDPI). DOI: 10.3390/math6120335 ISSN 2227-7390

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Official URL: https://www.mdpi.com/2227-7390/6/12/335

Abstract

In previous studies we used Laurent Schwartz’ theory of distributions to rigorously introduce discretizations and periodizations on tempered distributions. These results are now used in this study to derive a validity statement for four interlinking formulas. They are variants of Poisson’s Summation Formula and connect four commonly defined Fourier transforms to one another, the integral Fourier transform, the Discrete-Time Fourier Transform (DTFT), the Discrete Fourier Transform (DFT) and the Integral Fourier transform for periodic functions - used to analyze Fourier series. We prove that under certain conditions, these four Fourier transforms become particular cases of the Fourier transform in the tempered distributions sense. We first derive four interlinking formulas from four definitions of the Fourier transform pure symbolically. Then, using our previous results, we specify three conditions for the validity of these formulas in the tempered distributions sense.

Item URL in elib:https://elib.dlr.de/125155/
Document Type:Article
Title:Four Particular Cases of the Fourier Transform
Authors:
AuthorsInstitution or Email of AuthorsAuthors ORCID iD
Fischer, JensJens.Fischer (at) dlr.dehttps://orcid.org/0000-0002-8987-0859
Date:December 2018
Journal or Publication Title:Mathematics
Refereed publication:Yes
Open Access:Yes
Gold Open Access:Yes
In SCOPUS:Yes
In ISI Web of Science:No
Volume:6
DOI :10.3390/math6120335
Page Range:pp. 1-19
Editors:
EditorsEmail
UNSPECIFIEDMDPI
Publisher:Multidisciplinary Digital Publishing Institute (MDPI)
ISSN:2227-7390
Status:Published
Keywords:Fourier transform; Fourier series; DTFT; DFT; generalized functions; tempered distributions; Schwartz functions; Poisson Summation Formula; discretization; periodization
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:17 Dec 2018 08:38
Last Modified:21 Sep 2019 05:05

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