Preprint Article Version 1 This version is not peer-reviewed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)

How to cite: Fischer, J.V. There Is Only One Fourier Transform. Preprints 2017, 2017120173 (doi: 10.20944/preprints201712.0173.v1). Fischer, J.V. There Is Only One Fourier Transform. Preprints 2017, 2017120173 (doi: 10.20944/preprints201712.0173.v1).

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

Integral Fourier Transform; Discrete-Time Fourier Transform (DTFT); Discrete Fourier Transform (DFT); Integral Fourier Transform for periodic functions; Fourier series; Poisson Summation Formula; periodization trick; interpolation trick

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