# Hermite transform

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Hermite_transform
> Markdown URL: https://mediated.wiki/source/Hermite_transform.md
> Source: https://en.wikipedia.org/wiki/Hermite_transform
> Source revision: 1240093864
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In mathematics, the '''Hermite transform''' is an [integral transform](/source/integral_transform) named after the mathematician [Charles Hermite](/source/Charles_Hermite) that uses [Hermite polynomials](/source/Hermite_polynomials) <math>H_n(x)</math> as kernels of the transform.

The Hermite transform <math>H\{F(x)\} \equiv f_H (n)</math> of a function <math>F(x)</math> is
<math display="block">H\{F(x)\} \equiv f_H(n) = \int_{-\infty}^\infty e^{-x^2} \ H_n(x)\  F(x) \ dx</math>

The inverse Hermite transform <math>H^{-1}\{f_H(n)\}</math> is given by
<math display="block">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)</math> 

== Some Hermite transform pairs ==

{| class="wikitable" align="center"
!<math>F(x)\,</math>
!<math>f_H(n)\,</math>
|-
|<math>x^m</math>
|<math>\begin{cases} \frac{m!\sqrt{\pi} }{2^{m-n} \left(\frac{m-n}{2}\right)!}, & (m-n)\text{ even and} \geq0 \\
0, & \text{otherwise} \end{cases}</math><ref>{{Citation |last=McCully |first=Joseph Courtney |last2=Churchill |first2=Ruel Vance |date=1953 |title=Hermite and Laguerre integral transforms : preliminary report |url=http://deepblue.lib.umich.edu/handle/2027.42/6521 |language=en-US}}</ref>
|-
|<math>e^{ax}\,</math>
|<math>\sqrt\pi a^n e^{a^2/4}\,</math>
|-
|<math>e^{2xt-t^2}, \ |t|<\frac{1}{2}\,</math>
|<math>\sqrt\pi (2t)^n</math>
|-
|<math>H_m(x)\,</math>
|<math>\sqrt\pi 2^n n!\delta_{nm}\,</math>
|-
|<math>x^2H_m(x)\,</math>
|<math>2^n n! \sqrt{\pi}\begin{cases} 1 , & n=m+2 \\
\left(n+\frac{1}{2}\right), & n=m \\
(n+1)(n+2),& n=m-2 \\
0, & \text{otherwise}\end{cases}</math>
|-
|<math>e^{-x^2}H_m(x)\,</math>
|<math>\left(-1\right)^{p-m} 2^{p-1/2} \Gamma(p+1/2),\  m+n=2p,\  p\in\mathbb{Z}</math>
|-
|<math>H_m^2(x)\,</math>
|<math>\begin{cases}
 2^{m+n/2}\sqrt\pi \binom m{n/2}\frac{m!n!}{(n/2)!}, & n\text{ even and}\leq 2m \\
 0, & \text{otherwise}
\end{cases}</math><ref>{{cite journal |last=Feldheim |first=Ervin |date=1938 |title=Quelques nouvelles relations pour les polynomes d'Hermite |journal=Journal of the London Mathematical Society |language=fr |volume=s1-13 |pages=22–29 |doi=10.1112/jlms/s1-13.1.22}}</ref>
|-
|<math>H_m(x)H_p(x)\,</math>
|<math>\begin{cases}
    \frac{2^k\sqrt\pi m!n!p!}{(k-m)!(k-n)!(k-p)!} , &  n+m+p=2k,\ k\in\mathbb{Z};\ |m-p|\leq n\leq m+p\\
   0 ,              &  \text{otherwise}  
\end{cases}\,</math><ref>{{cite journal | last=Bailey | first=W. N. | title=On Hermite polynomials and associated Legendre functions | journal=Journal of the London Mathematical Society | issue=4 | date=1939 | pages=281–286 | volume=s1-14 | doi=10.1112/jlms/s1-14.4.281}}</ref>
|-
|<math>H_{n+p+q}(x)H_p(x)H_q(x)\,</math>
|<math>\sqrt\pi 2^{n+p+q} (n+p+q)!\,</math>
|-
|<math>\frac{d^m}{dx^m}F(x)\,</math>
|<math>f_H(n+m)\,</math>
|-
|<math>x\frac{d^m}{dx^m}F(x)\,</math>
|<math>nf_H(n+m-1)+\frac{1}{2}f_H(n+m+1)\,</math>
|-
|<math>e^{x^2}\frac{d}{dx}\left[e^{-x^2}\frac{d}{dx}F(x)\right]\,</math>
|<math>-2nf_H(n)\,</math>
|-
|<math>F(x - x_0)</math>
|<math>\sqrt{\pi}\sum^\infty_{k=0}\frac{(-x_0)^k}{k!}f_H(n+k)</math>
|-
|<math>F(x)*G(x)\,</math>
|<math>\sqrt\pi(-1)^n\left[2^{2n+1}\Gamma \left(n+\frac{3}{2}\right)\right]^{-1}f_H(n) g_H(n)\,</math><ref>{{cite journal | last=Glaeske | first=Hans-Jürgen | title=On a convolution structure of a generalized Hermite transformation | journal=Serdica Bulgariacae Mathematicae Publicationes | volume=9 | issue=2 | date=1983 | pages=223–229 | url=http://www.math.bas.bg/serdica/1983/1983-223-229.pdf}}</ref>
|-
|<math>e^{z^2} \sin(x z), \ |z|<\frac 12\ \,</math>
|<math>\begin{cases}
    \sqrt\pi (-1)^{\lfloor\frac{n}{2}\rfloor}(2z)^{n} , &  n\,\mathrm{odd}\\
   0 ,              &  n\,\mathrm{even}
\end{cases}\,</math>
|-
|<math>(1-z^2)^{-1/2} \exp\left[\frac{2xyz-(x^2+y^2)z^2}{(1-z^2)}\right]\,</math>
|<math>\sqrt\pi z^n H_n(y)</math><ref>{{harvnb|Erdélyi|Magnus|Oberhettinger|Tricomi|1955|page=194}}, 10.13 (22).</ref><ref>{{Citation |last1=Mehler |first1=F. G. |title=Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung |url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002152975 |journal=Journal für die Reine und Angewandte Mathematik |issue=66 |pages=161–176 |year=1866 |trans-title=On the development of a function of arbitrarily many variables according to higher-order Laplace functions |language=de |issn=0075-4102 |id={{ERAM|066.1720cj}}}}. See p. 174, eq. (18) and p. 173, eq. (13).</ref>
|-
|<math>\frac{H_m(y)H_{m+1}(x)-H_m(x)H_{m+1}(y)}{2^{m+1}m!(x-y)}</math>
|<math>\begin{cases}\sqrt{\pi}H_n(y) & n \leq m\\
0 & n > m
\end{cases}</math>
|}

==References==
{{Reflist}}

==Sources==
*{{citation |last1=Erdélyi |first1=Arthur |author-link1=Arthur Erdélyi |last2=Magnus |first2=Wilhelm |author-link2=Wilhelm Magnus |last3=Oberhettinger |first3=Fritz |author-link3=:de:Fritz Oberhettinger |last4=Tricomi |first4=Francesco G. |author-link4=Francesco Tricomi |title=Higher transcendental functions |volume=II |publisher=McGraw-Hill |year=1955 |url=http://apps.nrbook.com/bateman/Vol2.pdf |isbn=978-0-07-019546-2 |access-date=2023-11-09 |archive-date=2011-07-14 |archive-url=https://web.archive.org/web/20110714210423/http://apps.nrbook.com/bateman/Vol2.pdf |url-status=dead }}

Category:Integral transforms
Category:Mathematical physics

---
Adapted from the Wikipedia article [Hermite transform](https://en.wikipedia.org/wiki/Hermite_transform) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Hermite_transform?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.