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?
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$\begingroup$ Yes we can. But why do you need? $\endgroup$– user17762Mar 9, 2012 at 2:42
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5$\begingroup$ The key here was the phrase "fixed point" or "eigenfunction", then Google performed the rest of the work. mathoverflow.net/questions/12045/… and en.wikipedia.org/wiki/Fourier_transform#Eigenfunctions $\endgroup$– dlsMar 9, 2012 at 2:49
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$\begingroup$ Also, have a look at the answers here. $\endgroup$– kuch nahiMar 9, 2012 at 2:59
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2$\begingroup$ In 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, fis.unam.mx/~bwolf/Articles/28.pdf . You might follow the references in this paper, particularly those of Titchmarsh– Tom Copeland 1 min ago $\endgroup$– Tom CopelandApr 7, 2012 at 22:30
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$\begingroup$ See this MO post. $\endgroup$– NemoNov 24, 2017 at 10:02
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2 Answers
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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).
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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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$\begingroup$ Welcome 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. $\endgroup$– Tom-TomSep 29, 2015 at 19:38
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$\begingroup$ @Tom-Tom I'm a greenhorn here. Thanks for the suggestion. Will keep this in mind for all my future answers :) $\endgroup$ Jul 21, 2016 at 5:42