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Learned something new today, thank you! If I understand correctly, the Hermite functions are the eigenfunctions of the Fourier Transform and thus all have this
by mppm 1y ago
Learned something new today, thank you!
If I understand correctly, the Hermite functions are the eigenfunctions of the Fourier Transform and thus all have this property -- with the Gaussian being a special case. But sech(x) is doubly interesting because it is not a Hermite function, though it can be represented as an infinite series thereof. Are there other well-behaved examples of this, or is sech(x) unique in that regard?
- fxj 1y agoYes the dirac comb for example. Actually there are infinitely many. https://en.wikipedia.org/wiki/Dirac_comb https://en.wikipedia.org/wiki/Dirac_comb and for other: http://www.systems.caltech.edu/dsp/ppv/papers/journal08post/PPVIETEeigenFT.pdf http://www.systems.caltech.edu/dsp/ppv/papers/journal08post/...
- abetusk 1y agoA question I hadn't even thought to ask, thanks. So, basically, the eigenfunctions of the Fourier transform are Hermite polynomials times a Gaussian [0] [1]. [0] https://math.stackexchange.com/questions/728670/functions-that-are-their-own-fourier-transform https://math.stackexchange.com/questions/728670/functions-th... [1] https://en.wikipedia.org/wiki/Hermite_polynomials#Hermite_functions_as_eigenfunctions_of_the_Fourier_transform https://en.wikipedia.org/wiki/Hermite_polynomials#Hermite_fu...
- perihelions 1y agoAs well as the linear combinations (including infinite sums!) of Hermite functions with the same eigenvalue under the Fourier transform. (Those eigenvalues are infinitely degenerate). You could express sech(x) as such a sum.
- perihelions 1y agoThere has to be a link to the harmonic oscillator here. That's the Hamiltonian that's symmetric under exchange of position and momentum, and the Hermite functions are its eigenfunctions.
- kkylin 1y agoIndeed, the (quantum) harmonic oscillator Hamiltonian (with suitable scalings) commutes with the Fourier transform. Since the former has the Hermite functions as eigenbasis, the Hermite functions also form an eigenbasis for the latter.
- deleted 1y ago[deleted]