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There Is Only One Fourier Transform

A peer-reviewed version of this preprint was published in:
Mathematics 2018, 6(12), 335. https://doi.org/10.3390/math6120335

Submitted:

24 December 2017

Posted:

25 December 2017

Read the latest preprint version here

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: 
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
Subject: 
Computer Science and Mathematics  -   Analysis
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.