Editing Hermite polynomials (section)

{{Short description|Polynomial sequence}}
{{About|the family of orthogonal polynomials on the real line|polynomial interpolation on a segment using derivatives|Hermite interpolation|integral transform of Hermite polynomials|Hermite transform}}
{{Use American English|date = March 2019}}

In [[mathematics]], the '''Hermite polynomials''' are a classical [[orthogonal polynomials|orthogonal]] [[polynomial sequence]].

The polynomials arise in:
* [[signal processing]] as [[Hermitian wavelet]]s for [[wavelet transform]] analysis
* [[probability]], such as the [[Edgeworth series]], as well as in connection with [[Brownian motion]];
* [[combinatorics]], as an example of an [[Appell sequence]], obeying the [[umbral calculus]];
* [[numerical analysis]] as [[Gaussian quadrature]]; 
* [[physics]], where they give rise to the [[eigenstate]]s of the [[quantum harmonic oscillator]]; and they also occur in some cases of the [[heat equation]] (when the term <math>\begin{align}xu_{x}\end{align}</math> is present);
* [[systems theory]] in connection with nonlinear operations on [[Gaussian noise]]. 
* [[random matrix theory]] in [[Gaussian ensemble]]s.

Hermite polynomials were defined by [[Pierre-Simon Laplace]] in 1810,<ref>{{cite journal |last1=Laplace |title=Mémoire sur les intégrales définies et leur application aux probabilités, et spécialement a la recherche du milieu qu'il faut choisir entre les resultats des observations |journal=Mémoires de la Classe des Sciences Mathématiques et Physiques de l'Institut Impérial de France |date=1811 |volume=11 |pages=297–347 |url=https://www.biodiversitylibrary.org/item/55081#page/293/mode/1up |trans-title=Memoire on definite integrals and their application to probabilities, and especially to the search for the mean which must be chosen among the results of observations |language=French}}</ref><ref>{{Citation |first=P.-S. |last=Laplace |title=Théorie analytique des probabilités |trans-title=Analytic Probability Theory |date=1812 |volume=2 |pages=194–203}} Collected in [https://gallica.bnf.fr/ark:/12148/bpt6k775950.r=Oeuvres%20complètes%20de%20Laplace.%20Tome%207?rk=21459;2 ''Œuvres complètes'' '''VII'''].</ref> though in scarcely recognizable form, and studied in detail by [[Pafnuty Chebyshev]] in 1859.<ref>{{cite journal |first=P. |last=Tchébychef |title=Sur le développement des fonctions à une seule variable |trans-title=On the development of single-variable functions |journal=Bulletin de l'Académie impériale des sciences de St.-Pétersbourg |volume=1 |date=1860 |pages=193–200 |url=https://www.biodiversitylibrary.org/item/104584#page/129/mode/1up |language=French }} Collected in [https://archive.org/details/117744684_001/page/n511/mode/2up ''Œuvres'' '''I''', 501–508.]</ref> Chebyshev's work was overlooked, and they were named later after [[Charles Hermite]], who wrote on the polynomials in 1864, describing them as new.<ref>{{cite journal |first=C. |last=Hermite |title=Sur un nouveau développement en série de fonctions |trans-title=On a new development in function series |journal=C. R. Acad. Sci. Paris |volume=58 |date=1864 |pages=93–100, 266–273 |url=https://www.biodiversitylibrary.org/item/23663#page/99/mode/1up |language=French }} Collected in ''Œuvres'' '''II''', 293–308.</ref> They were consequently not new, although Hermite was the first to define the multidimensional polynomials.