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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fourier-analysis
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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