Competing definitions of the Fourier transform

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Discussion Overview

The discussion revolves around the different definitions of the Fourier transform, particularly focusing on the presence of the factor of \(2\pi\) in the definitions and its implications for normalization and equivalence. Participants explore theoretical aspects of the Fourier transform, including its inverse and the conventions used in various texts.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes two definitions of the Fourier transform, questioning their equivalence and the significance of the \(2\pi\) factor.
  • Another participant suggests consulting a specific text for clarification on the definitions.
  • Some participants argue that the definitions are equivalent, with the factor of \(1/\sqrt{2\pi}\) included to ensure proper normalization when applying the transform and its inverse.
  • It is mentioned that omitting the \(2\pi\) from the definition can lead to complications in the inverse transform, emphasizing the need to adhere to specific conventions in different texts.
  • A later reply proposes a method to verify the equivalence of the definitions by making a change of variables and checking the resulting coefficients in the transforms.

Areas of Agreement / Disagreement

Participants express differing views on the equivalence of the definitions and the necessity of the \(2\pi\) factor, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

There are limitations regarding the assumptions made about the definitions and the conventions used in various texts, which are not universally agreed upon. The discussion also highlights potential confusion in the application of the Fourier transform and its inverse.

AxiomOfChoice
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Just began a serious study of the Fourier transform with a couple of books. One of them defines the Fourier transform on [itex]\mathbb R[/itex] as

[tex] \hat f(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(x) e^{-i\xi x}dx.[/tex]

Another defines it as

[tex] \hat f(\xi) = \int_{-\infty}^\infty f(x) e^{-2\pi i \xi x} dx.[/tex]

A few questions:

(1) Are these definitions somehow equivalent? I cannot seem to obtain the second from the first by making a simple change of variables.

(2) Why worry about the factors of [itex]2\pi[/itex] in the definitions? What does that do for us? Why not leave those out altogether?
 
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See M.L.Boas - "Mathematical Methods" for disambiguation. I had the same problem and it really helped me.
 
The definitions are equivalent, and the factor of [itex]1/\sqrt{2\pi}[/itex] is added into ensure that applying the transform and its inverse doesn't mulitply the result by a constant factor other than 1. You need to know whether the [itex]2\pi[/itex] is in the exponential or not to figure out this normalization constant.
 
AxiomOfChoice said:
(2) Why worry about the factors of [itex]2\pi[/itex] in the definitions? What does that do for us? Why not leave those out altogether?

If you leave them out of the definition of the Fourier transform, they appear in the inverse transform. If you try to leave them all out, then IFT ( FT [ f(x) ] ) doesn't equal f(x).

Unfortunately there isn't an "obvious" place to put the ##2\pi## that everybody agrees on, so you have to check what convention any particular book or paper is using.

Similar issues apply to the discrete Fourier transform, and computer software routines that calculate it. There you also have to watch out for factors of n and 1/n, where n is the number of samples in the DFT.
 
Oh. that is easily solved. Make change of variables in the familiar transform you understand clearly to include the 2[itex]\pi[/itex] in the exponent, THEN see if the product of coefficients of Fourier and inverse Fourier transforms gives the factor claimed. if they are the same, this is correct, if it is not, it got to be wrong.
 

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