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Open AccessArticle

Four Particular Cases of the Fourier Transform

German Aerospace Center (DLR), Microwaves and Radar Institute, 82234 Wessling, Germany
Mathematics 2018, 6(12), 335; https://doi.org/10.3390/math6120335
Received: 8 October 2018 / Revised: 29 November 2018 / Accepted: 14 December 2018 / Published: 18 December 2018
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. View Full-Text
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 Fourier transform; Fourier series; Discrete-Time Fourier Transform (DTFT); Discrete Fourier Transform (DFT); generalized functions; tempered distributions; Schwartz functions; Poisson Summation Formula; discretization; periodization
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MDPI and ACS Style

Fischer, J.V. Four Particular Cases of the Fourier Transform. Mathematics 2018, 6, 335.

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