Four Particular Cases of the Fourier Transform

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

Mathematics, 2018, vol. 6, issue 12, 1-19

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.

Keywords: Fourier transform; Fourier series; Discrete-Time Fourier Transform (DTFT); Discrete Fourier Transform (DFT); generalized functions; tempered distributions; Schwartz functions; Poisson Summation Formula; discretization; periodization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2018
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
https://www.mdpi.com/2227-7390/6/12/335/pdf (application/pdf)
https://www.mdpi.com/2227-7390/6/12/335/ (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:6:y:2018:i:12:p:335-:d:191376

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 ().