Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed

Four Particular Cases of the 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. Four Particular Cases of the Fourier Transform. Preprints 2017, 2017120173. https://doi.org/10.20944/preprints201712.0173.v3 Fischer, J.V. Four Particular Cases of the Fourier Transform. Preprints 2017, 2017120173. https://doi.org/10.20944/preprints201712.0173.v3

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; DTFT; DFT; generalized functions; tempered distributions; Schwartz functions; Poisson Summation Formula; discretization; periodization

Subject

Computer Science and Mathematics, Analysis

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