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I am currently writing my PhD and would like to display a table summing up the different kind of Fourier Transforms and Fourier Series. Here is the table I got from my different readings (mostly Oppenheim, A. V., Schafer, R. W., & Buck, J. R. (1999). Discrete-time signal processing and Percival, D. B., & Walden, A. T. (1998). Spectral analysis for physical applications):

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Is this table right?

Gim
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    If I answer, do I get mentioned in the thesis? – Chester Sep 09 '15 at 20:15
  • For periodic, continuous input, continuous output, the Fourier transform is different from the Fourier transform in the non-periodic, continuous input case, because what is typically called the "Fourier transform" is defined on $L^1$ functions. – Christopher A. Wong Sep 09 '15 at 20:19
  • @Chester Sure! And if you give me your email I'll even send you a pdf so you'll be able to check...:) – Gim Sep 09 '15 at 20:23
  • @ChristopherA.Wong Are they both called the same way? There is also a mention of this in this post, i.e. the Fourier transform can be applied on one period only of the continuous periodic function – Gim Sep 09 '15 at 20:27
  • I don't know whether they are called the same thing, but I do know they are not the same thing. Unfortunately I think mathematicians and engineers generally have disagreements over terminology and notation. In truth, everything in your table is a type of Fourier transform. – Christopher A. Wong Sep 09 '15 at 20:32

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A very nice article about the different kind of Fourier transforms as well as their interpretation is available on the Gaussian Waves website : http://www.gaussianwaves.com/2015/11/interpreting-fft-results-complex-dft-frequency-bins-and-fftshift/

Gim
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  • you know that the Fourier transform of tempered distributions generalizes all this ? it let us understand what happens when the sample rate or the periodicity tends to $\infty$ and how one transform is the limiting case of another (so that all the Fourier transforms are limiting cases of the discrete Fourier transform) – reuns Feb 14 '16 at 21:30