Гермит полиномдары: қасиеттері мен қолданыстары

Polynomial sequence
the family of orthogonal polynomials on the real line

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, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, describing them as new. They were consequently not new, although Hermite was the first to define the multidimensional polynomials.

The nth order Hermite polynomial is a polynomial of degree n. The probabilist's version Hen has leading coefficient 1, while the physicist's version Hn has leading coefficient 2^(n).

Similar to Taylor expansion, some functions are expressible as an infinite sum of Hermite polynomials. Specifically, if , then it has an expansion in the physicist's Hermite polynomials. Given such , the partial sums of the Hermite expansion of converges to in the norm if and only if .

The probabilist's Hermite polynomials satisfy the identity where D represents differentiation with respect to x, and the exponential is interpreted by expanding it as a power series. There are no delicate questions of convergence of this series when it operates on polynomials, since all but finitely many terms vanish. Since the power series coefficients of the exponential are well known, and higher order derivatives of the monomial x^(n) can be written down explicitly, this differential operator representation gives rise to a concrete formula for the coefficients of Hn that can be used to quickly compute these polynomials. Since the formal expression for the Weierstrass transform W is e^(D^(2)), we see that the Weierstrass transform of is x^(n). Essentially the Weierstrass transform thus turns a series of Hermite polynomials into a corresponding Maclaurin series. The existence of some formal power series g(D) with nonzero constant coefficient, such that 1=Hen(x) = g(D)x^(n), is another equivalent to the statement that these polynomials form an Appell sequence. Since they are an Appell sequence, they are a fortiori a Sheffer sequence.