Hermite transform

In mathematics, the Hermite transform is an integral transform named after the mathematician Charles Hermite that uses Hermite polynomials ${\displaystyle H_{n}(x)}$ as kernels of the transform.

The Hermite transform ${\displaystyle H\{F(x)\}\equiv f_{H}(n)}$ of a function ${\displaystyle F(x)}$ is $${\displaystyle H\{F(x)\}\equiv f_{H}(n)=\int _{-\infty }^{\infty }e^{-x^{2}}\ H_{n}(x)\ F(x)\ dx}$$

The inverse Hermite transform ${\displaystyle H^{-1}\{f_{H}(n)\}}$ is given by $${\displaystyle H^{-1}\{f_{H}(n)\}\equiv F(x)=\sum _{n=0}^{\infty }{\frac {1}{{\sqrt {\pi }}2^{n}n!}}f_{H}(n)H_{n}(x)}$$