properties of orthogonal polynomials

A countable system of orthogonal polynomials

p0(x),p1(x),p2(x), (1)

on an interval  [a,b],  where a inner product of two functionsMathworldPlanetmath


is defined with respect to a weighting function W(x), satisfies the orthogonality condition (

(pm,pn)= 0always whenmn.

One also requires that

deg(pn(x))=nfor all n.

Such a system (1) may be used as basis for the vector spaceMathworldPlanetmath of functions defined on  [a,b], i.e. certain such functions f may be expanded as a series (


where the coefficients cn have the expression


Other properties

  • The basis property of the system (1) comprises that any polynomial P(x) of degree n can be uniquely expressed as a finite linear combinationMathworldPlanetmath

  • Every member pn(x) of (1) is orthogonalMathworldPlanetmath to any polynomial P(x) of degree less than n.

  • There is a recurrence relation


    enabling to determine a .

  • The zeros of pn(x) are all real and belong to the open interval(a,b);  between two of those zeros there are always zeros of pn+1(x).

  • The Sturm–Liouville differential equationMathworldPlanetmath

    Q(x)p′′+L(x)p+λp= 0, (2)

    where Q(x) is a polynomial of at most degree 2 and L(x) a linear polynomial, gives under certain conditions, as p a system of orthogonal polynomials p0,p1,  corresponding suitable values (eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath) λ0,λ1,  of the parametre λ.  Those satisfy the Rodrigues formulaPlanetmathPlanetmath


    where kn is a constant and


    The classical Chebyshev (, Hermite (, Laguerre (, and Legendre polynomialsDlmfDlmfMathworld all satisfy an equation (2).

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Title properties of orthogonal polynomials
Canonical name PropertiesOfOrthogonalPolynomials
Date of creation 2013-03-22 19:05:34
Last modified on 2013-03-22 19:05:34
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Topic
Classification msc 42C05
Classification msc 33D45
Related topic HilbertSpace
Related topic TopicsOnPolynomials
Related topic IndexOfSpecialFunctions
Related topic OrthogonalityOfLaguerrePolynomials
Related topic OrthogonalityOfChebyshevPolynomials
Defines Rodrigues formula