原文地址
傅立叶变换之后还是自己的函数都有哪些？ - yhm138的回答 - 知乎
https://www.zhihu.com/question/460357978/answer/1908160414
太长不看
a Gaussian function times a Hermite polynomial
DiracComb也行
正文
里面提到，"There is a whole class of them that satisfy this eigenvalue condition of \(Ff=cf\) .They are essentially a Gaussian function times a Hermite polynomial"
使用Mathematica验证这个等式 \(\mathcal{F}\left\{e^{-\frac{1}{2} x^2} H_n(x)\right\}=(-i)^n e^{-\frac{1}{2} k^2} H_n(k)\)的前几项
Clear["Global`*"]
isEqual[n_Integer] :=
FourierTransform[Exp[-x^2/2]*HermiteH[n, x], x, k, FourierParameters -> {0, -1}]
==
(-I)^n*Exp[-k^2/2]*HermiteH[n, k] // FullSimplify
isEqual[#] & /@ Range[20]
DiracComb也是可以的
