orthogonal polynomials

1 Orthogonal Polynomials

Polynomials of degree n are analytic functionsMathworldPlanetmath that can be written in the form


They can be differentiated and integrated for any value of x, and are fully determined by the n+1 coefficients a0an . For this simplicity they are frequently used to approximate more complicated or unknown functionsMathworldPlanetmath. In approximations, the necessary degree n of the polynomial is not normally defined by criteria other than the quality of the approximation.

Using polynomials as defined above tends to lead into numerical difficulties when determining the ai, even for small values of n. It is therefore customary to stabilize results numerically by using orthogonal polynomials over an interval [a,b], defined with respect to a positive weighting function W(x)>0 by


Orthogonal polynomials are obtained in the following way: define the scalar product.


between the functions f and g, where W(x) is a weight factor. Starting with the polynomials p0(x)=1, p1(x)=x, p2(x)=x2, etc., from the Gram-Schmidt decomposition one obtains a sequence of orthogonal polynomials q0(x),q1(x),, such that (qm,qn)=Nnδmn. The normalization factors Nn are arbitrary. When all Ni are equal to one, the polynomials are called orthonormal.

Some important orthogonal polynomials are:

a b W(x) name
-1 1 1 Legendre polynomialsDlmfDlmfMathworldPlanetmath
-1 1 (1-x2)-1/2 Chebyshev polynomials
- e-x2 Hermite polynomialsMathworldPlanetmath

Orthogonal polynomials of successive orders can be expressed by a recurrence relation


This relation can be used to compute a finite series


with arbitrary coefficients ai, without computing explicitly every polynomial pj (Horner’s Rule).

Chebyshev polynomials Tn(x) are also orthogonal with respect to discrete values xi:


where the xi depend on M.

For more information, see [2, 3].


  • 1 Originally from The Data Analysis Briefbook (http://rkb.home.cern.ch/rkb/titleA.htmlhttp://rkb.home.cern.ch/rkb/titleA.html)
  • 2 M. Abramowitz and I.A. Stegun (Eds.), Handbook of Mathematical Functions, National Bureau of Standards, Dover, New York, 1974.
  • 3 W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C, Second edition, Cambridge University Press, 1995. (The same book exists for the Fortran language). There is also an Internet version which you can work from.
Title orthogonal polynomials
Canonical name OrthogonalPolynomials
Date of creation 2013-03-22 12:06:20
Last modified on 2013-03-22 12:06:20
Owner akrowne (2)
Last modified by akrowne (2)
Numerical id 11
Author akrowne (2)
Entry type Definition
Classification msc 42C05
Classification msc 33D45
Related topic OrthogonalMatrices
Related topic Orthonormal