Preprint Article Version 2 This version is not peer-reviewed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)

How to cite: Fischer, J.V. There Is Only One Fourier Transform. Preprints 2017, 2017120173 (doi: 10.20944/preprints201712.0173.v2). Fischer, J.V. There Is Only One Fourier Transform. Preprints 2017, 2017120173 (doi: 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.

Subject Areas

Fourier Transform; Discrete-Time Fourier Transform (DTFT); Discrete Fourier Transform (DFT); Fourier Series; Poisson Summation Formula; discretization; periodization; discrete function; periodic function; periodization trick

Comments (3)

Comment 1
Received: 23 May 2018
Commenter: Jens V. Fischer (Click to see Publons profile: )
The commenter has declared there is no conflict of interests.
Comment: CAUTION: An unauthorized copy of this paper (awfully edited, very confusing and peppered with many mistakes) may be found on predatory open-access platforms. Please do not cite them. Instead, you may cite this preprint via its document identifier (DOI) - it will always be linked to a latested version of this paper.

A list of predatory publishers can be found here:
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Comment 2
Received: 25 May 2018
Commenter: Jens V. Fischer (Click to see Publons profile: )
The commenter has declared there is no conflict of interests.
Comment: Please always refer to the latest Version of this paper at https://www.preprints.org/manuscript/201712.0173/ .
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Comment 3
Received: 25 May 2018
Commenter: Jens V. Fischer (Click to see Publons profile: )
The commenter has declared there is no conflict of interests.
Comment: Please feel invited to do a publicly visible review of this paper at https://publons.com/publon/2333534/ .
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