We know that the Fourier transform of a Gaussian function is Gaussian function itself. Can anyone give one or more functions which have themselves as Fourier transform?
Which function's Fourier transform is the function itself?
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0Yes we can. But why do you need? – 2012-03-09
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5The key here was the phrase "fixed point" or "eigenfunction", then Google performed the rest of the work. http://mathoverflow.net/questions/12045/what-are-fixed-points-of-the-fourier-transform and http://en.wikipedia.org/wiki/Fourier_transform#Eigenfunctions – 2012-03-09
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0Also, have a look at the answers [here](http://math.stackexchange.com/questions/10774/how-do-i-compute-the-eigenfunctions-of-the-fourier-transform). – 2012-03-09
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2In the old days the "fixed-point functions" were called self-reciprocal functions and investigated by the likes of Hardy and Titchmarsh. See "On self-reciprocal functions under a class of integral transforms" by Kurt Wolf, http://www.fis.unam.mx/~bwolf/Articles/28.pdf . You might follow the references in this paper, particularly those of Titchmarsh– Tom Copeland 1 min ago – 2012-04-07
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0See this [MO post](https://mathoverflow.net/a/224201/82588). – 2017-11-24
2 Answers
Well, the Dirac delta function, strictly speaking, is not a 'function' but it serves your purpose.
The Fourier transform of a Dirac comb (or an impulse train, as it is called in Electrical Engineering) is another Dirac comb.
$$ \sum_{n=-\infty}^{\infty} \delta(t-nT_o) = \frac {2\pi} {T_o} \sum_{m=-\infty}^{\infty} \delta(\omega - m\omega_o) $$
For the proof, refer http://nptel.ac.in/courses/IIT-MADRAS/Principles_Of_Communication/pdf/Lecture07_FTPeriodic.pdf
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0Welcome to Maths.SE ! Thanks for your answer. Please note that on this site it is recomended to include excerpts of your work, providing a link is not enough to reach this site's standards. – 2015-09-29
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0@Tom-Tom I'm a greenhorn here. Thanks for the suggestion. Will keep this in mind for all my future answers :) – 2016-07-21
Pick a function $f$ that is reasonable enough for the inversion formula to hold (e.g. take $f$ in Schwartz space, which contains the Gaussian among other functions). If $\mathcal{F}$ denotes the linear transformation which takes $f$ to its Fourier transform, then it's easy to check that $\mathcal{F}^{4}$ is the identity map. In particular, by playing some games you find that $$g \ = \ f + \mathcal{F}(f) + \mathcal{F}^{2}(f) + \mathcal{F}^{3}(f) $$ is fixed by $\mathcal{F}$. So $g$ is its own Fourier transform.
This argument doesn't produce a concrete function, but it at least shows you that the Gaussian is far from the only function that is equal to its own Fourier transform. If you want a more specific example, you can show that $(\cosh \pi x)^{-1}$ is its own Fourier transform (use contour integration and the residue theorem).