Orthogonal Polynomial

orthogonal polynomial

[ȯr′thäg·ən·əl päl·ə′nō·mē·əl]
(mathematics)
Orthogonal polynomials are various families of polynomials, which arise as solutions to differential equations related to the hypergeometric equation, and which are mutually orthogonal as functions.

Orthogonal Polynomial

 

A set of orthogonal polynomials is a special system of polynomials {pn(x)}, n = 0, 1, 2, …, that are orthogonal with respect to some weight function ρ(x) on an interval [a, b]. A normalized system of orthogonal polynomials is denoted by n(x), and a system of orthogonal polynomials whose leading coefficients are all equal to 1 is denoted by n (x). In boundary value problems of mathematical physics, systems of orthogonal polynomials are often encountered where the weight function ρ(x) satisfies the following differential equation (Pearson’s equation):

A polynomial pn (x) in such a system satisfies the differential equation

where λn = n[α1 + (n + 1) β2].

The most important classical systems of orthogonal polynomials are of this type. Aside from constant multipliers, the polynomials are determined by the choices of a, b, and ρ(x) indicated below.

(1) Jacobi polynomials, {pn(λ,μ)(x)}, correspond to a = –1, b = 1, ρ(x) = (1 – x)λ (1 + x)μ, λ > –1, and μ > –1. Certain values of λ and μ are associated with special cases of Jacobi polynomials. Thus, for λ = μ we have the ultraspherical polynomials, Pn(λ)(x), which are sometimes called Gegen-bauer polynomials. When λ = μ = –½, that is, ρ(x) = Orthogonal Polynomial we have the Chebyshev polynomials of the first kind, Tn(x); and for λ = μ = ½, that is Orthogonal Polynomial, we have the Chebyshev polynomials of the second kind, Un(x). For λ = μ = 0, that is, ρ(x) ≡ 1, we have the Legendre polynomials, Pn(x).

(2) Laguerre polynomials, Ln (x), correspond to a = 0, b = + ∞, and ρ(x) = e–x. For ρ(x) = xae–x >–1), we have the generalized Laguerre polynomials, Orthogonal Polynomial.

(3) Hermite polynomials, Hn(x), for a = –, ∞, b = + ∞, and

ρ(x) = ex2

Orthogonal polynomials have many properties in common. The zeroes of the polynomials pn(x) are real and simple and are located within the interval [a, b]. Between any two successive zeroes of pn (x) there lies a zero of pn+1 (x). The polynomial pn (x) is given by Rodrigues’ formula

where An is a constant and β(x) is of the form given in (*) above. Our systems of orthogonal polynomials are complete. Three successive orthogonal polynomials n(x), n + 1 (x), and n + 2 (x) are related by the recursion formula

n + 2 (x) = (xαn + 2) n + 1)(xλn + 1(x)

where αn + 2 and λn + 1 are given by the coefficients of the polynomials as follows: if

then

αn + 2 = αn + 1,nαn + 2, n + 1

λn + 1 = αn + 1, n – 1αn + 2αn + 1, nαn + 2, n

P. L. Chebyshev developed a general theory of orthogonal polynomials. His basic method of studying orthogonal polynomials was to expand the integral

in a continued fraction with partial denominators xαn and partial numerators λn – 1. The denominators of the convergents ϕn(x)/pn (x) of this continued fraction form a system of orthogonal polynomials on the interval [a, b] with respect to the weight function ρ(x).

The classical systems of orthogonal polynomials given above can be expressed in terms of the hypergeometric function.

REFERENCE

Szegö, G. Ortogonal’nye mnogochleny. Moscow, 1962. (Translated from English.)

V. I. BITIUTSKOV

References in periodicals archive ?
27 Comparison Protein source ns * R:C ratio ** ** Interaction Ns ns Orthogonal polynomial R:C (linear) ** ** R:C (quadratic) ns ** R:C (cubic) ns ns Protein source (1) R:[C.
3], q)}n generalizes the orthogonal polynomial system associated with X [4, 13].
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator.
In this paper, we used multiple orthogonal polynomial regression design to process the experimental data, considered the interaction of two factors.
Among the 12 mathematical topics are a survey of quadrature rules on the unit circle, the multilineal trigonometric problems of moments, multiple orthogonal polynomial ensembles, greedy energy points with external fields, relative asymptotics for general orthogonal polynomials, a progress report on the fine structure of zeros of orthogonal polynomials, and some open problems.
In [section] 4 the Gegenbauer reconstruction method is first reviewed for one-dimensional problems using both Fourier and orthogonal polynomial expansions.
An orthogonal polynomial model was used to fit the frequency sweep curves for analyzing the elastic viscous moduli of creme fraiche.
is called an orthogonal polynomial sequence with respect to the measure w if it satisfies the following two conditions:
7] Frank Filbir, Roland Girgensohn, Anu Saxena, Ajit Iqbal Singh, Ryszard Szwarc, Simultaneous Preservation of Orthogonality of Polynomials by Linear Operators Arising from Dilation of Orthogonal Polynomial Systems, Journal of Computational Analysis and Applications, 2(2000), 177-213.
The corresponding (monic) orthogonal polynomial [[PI].
n] be the roots of the classical orthogonal polynomial [q.
2])} be the monic orthogonal polynomial sequences on T associated to [mu] and [[mu].

Full browser ?