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 $$H^{-1}\{f_H(n)\} = F(x) = \sum_{n=0}^\infty \frac{1}{\sqrt\pi 2^n n!} f_H(n) H_n(x)$$