In mathematics, the Hermite transform is an integral transform named after the mathematician Charles Hermite that uses Hermite polynomials H n ( x ) {\displaystyle H_{n}(x)} as kernels of the transform.
The Hermite transform H { F ( x ) } ≡ f H ( n ) {\displaystyle H\{F(x)\}\equiv f_{H}(n)} of a function F ( x ) {\displaystyle F(x)} is H { F ( x ) } ≡ f H ( n ) = ∫ − ∞ ∞ e − x 2 H n ( x ) F ( x ) d x {\displaystyle H\{F(x)\}\equiv f_{H}(n)=\int _{-\infty }^{\infty }e^{-x^{2}}\ H_{n}(x)\ F(x)\ dx}
The inverse Hermite transform H − 1 { f H ( n ) } {\displaystyle H^{-1}\{f_{H}(n)\}} is given by H − 1 { f H ( n ) } ≡ F ( x ) = ∑ n = 0 ∞ 1 π 2 n n ! f H ( n ) H n ( x ) {\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)}