Hermite polynomials
Polynomial sequence / From Wikipedia, the free encyclopedia
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This article is about the family of orthogonal polynomials on the real line. For polynomial interpolation on a segment using derivatives, see Hermite interpolation. For integral transform of Hermite polynomials, see Hermite transform.
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:
- signal processing as Hermitian wavelets 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 eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term is present);
- systems theory in connection with nonlinear operations on Gaussian noise.
- random matrix theory in Gaussian ensembles.
Hermite polynomials were defined by Pierre-Simon Laplace in 1810,[1][2] though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859.[3] Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, describing them as new.[4] They were consequently not new, although Hermite was the first to define the multidimensional polynomials.