In mathematics, the '''Hermite transform''' is an integral transform named after the mathematician Charles Hermite that uses 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