# Hermite transform

The integral

$$f ( n) = H \{ F ( x) \} = \int\limits _ {- \infty } ^ \infty e ^ {- x ^ {2} } H _ {n} ( x) F ( x) d x ,\ \ n = 0 , 1 \dots$$

where $H _ {n} ( x)$ are the Hermite polynomials. The inversion formula is

$$F ( x) = \sum _ { n= 0} ^ \infty \frac{1}{\sqrt \pi } \frac{f ( n) }{2 ^ {n} n ! } H _ {n} ( x) = \ H ^ {-1} \{ f ( n) \} ,\ - \infty < x < \infty ,$$

provided that the series converges. The Hermite transform reduces the operator

$$R [ F ( x) ] = e ^ {x ^ {2} } \frac{d}{dx} \left [ e ^ {x ^ {2} } \frac{d}{dx} F ( x) \right ]$$

to an algebraic one by the formula

$$H \{ R [ F ( x) ] \} = - 2 n f ( n) .$$

If $F$ and all its derivatives up to and including the $p$- th order are bounded, then

$$H \{ F ^ { ( p) } ( x) \} = f ( n + p ) .$$

The Hermite transform has also been introduced for a special class of generalized functions (see ). They are used to solve differential equations containing the operator $R$.

How to Cite This Entry:
Hermite transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermite_transform&oldid=51095
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article