Hermite transform

In mathematics, Hermite transform is an integral transform named after the mathematician Charles Hermite, which uses Hermite polynomials $H_{n}(x)$ as kernels of the transform. This was first introduced by Lokenath Debnath in 1964.

The Hermite transform of a function $F(x)$ is

H\{F(x)\}=f_{H}(n)=\int _{-\infty }^{\infty }e^{-x^{2}}\ H_{n}(x)\ F(x)\ dx

The inverse Hermite transform is given by

H^{-1}\{f_{H}(n)\}=F(x)=\sum _{n=0}^{\infty }{\frac {1}{{\sqrt {\pi }}2^{n}n!}}f_{H}(n)H_{n}(x)

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