DLR-Logo -> http://www.dlr.de
DLR Portal Home | Imprint | Privacy Policy | Contact | Deutsch
Fontsize: [-] Text [+]

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.

[img] PDF - Published version

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


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
AuthorsInstitution or Email of AuthorsAuthor's ORCID iDORCID Put Code
Fischer, JensUNSPECIFIEDhttps://orcid.org/0000-0002-8987-0859UNSPECIFIED
Date:December 2018
Journal or Publication Title:Mathematics
Refereed publication:Yes
Open Access:Yes
Gold Open Access:Yes
In ISI Web of Science:Yes
Page Range:pp. 1-19
EditorsEmailEditor's ORCID iDORCID Put Code
Publisher:Multidisciplinary Digital Publishing Institute (MDPI)
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 - 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:17 Dec 2018 08:38
Last Modified:30 Oct 2023 14:28

Repository Staff Only: item control page

Help & Contact
electronic library is running on EPrints 3.3.12
Website and database design: Copyright © German Aerospace Center (DLR). All rights reserved.