Article
Version 2
Preserved in Portico This version is not peer-reviewed
There Is Only One Fourier Transform
Version 1
: Received: 24 December 2017 / Approved: 25 December 2017 / Online: 25 December 2017 (09:00:18 CET)
Version 2 : Received: 22 May 2018 / Approved: 23 May 2018 / Online: 23 May 2018 (07:42:15 CEST)
Version 3 : Received: 8 October 2018 / Approved: 9 October 2018 / Online: 9 October 2018 (05:40:13 CEST)
Version 4 : Received: 21 November 2018 / Approved: 21 November 2018 / Online: 21 November 2018 (11:09:28 CET)
Version 2 : Received: 22 May 2018 / Approved: 23 May 2018 / Online: 23 May 2018 (07:42:15 CEST)
Version 3 : Received: 8 October 2018 / Approved: 9 October 2018 / Online: 9 October 2018 (05:40:13 CEST)
Version 4 : Received: 21 November 2018 / Approved: 21 November 2018 / Online: 21 November 2018 (11:09:28 CET)
How to cite: Fischer, J.V. There Is Only One Fourier Transform. Preprints 2017, 2017120173. https://doi.org/10.20944/preprints201712.0173.v2 Fischer, J.V. There Is Only One Fourier Transform. Preprints 2017, 2017120173. https://doi.org/10.20944/preprints201712.0173.v2
Abstract
Four Fourier transforms are usually defined, the Integral Fourier transform, the Discrete-Time Fourier transform (DTFT), the Discrete Fourier transform (DFT) and the Integral Fourier transform for periodic functions. However, starting from their definitions, we show that all four Fourier transforms can be reduced to actually only one Fourier transform, the Fourier transform in the distributional sense.
Keywords
Fourier Transform; Discrete-Time Fourier Transform (DTFT); Discrete Fourier Transform (DFT); Fourier Series; Poisson Summation Formula; discretization; periodization; discrete function; periodic function; periodization trick
Subject
Computer Science and Mathematics, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Comments (3)
We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.
Leave a public commentSend a private comment to the author(s)
* All users must log in before leaving a comment
Commenter:
The commenter has declared there is no conflict of interests.
A list of predatory publishers can be found here:
https://predatoryjournals.com/publishers .
More about predatory Open Access publishing can be found here:
https://en.wikipedia.org/wiki/Predatory_open_access_publishing .
Commenter:
The commenter has declared there is no conflict of interests.
Commenter:
The commenter has declared there is no conflict of interests.