properties of orthogonal polynomials

A countable system of orthogonal polynomials

 $\displaystyle p_{0}(x),\,p_{1}(x),\,p_{2}(x),\,\ldots$ (1)

on an interval  $[a,\,b]$,  where a inner product of two functions  $(f,\,g)\;:=\;\int_{a}^{b}\!f(x)g(x)W(x)\,dx$

is defined with respect to a weighting function $W(x)$, satisfies the orthogonality condition (http://planetmath.org/OrthogonalVectors)

 $(p_{m},\,p_{n})\;=\;0\quad\mbox{always when}\quad m\neq n.$

One also requires that

 $\deg\left(p_{n}(x)\right)\;=\;n\quad\mbox{for all }n.$

Such a system (1) may be used as basis for the vector space  of functions defined on  $[a,\,b]$, i.e. certain such functions $f$ may be expanded as a series (http://planetmath.org/FunctionSeries)

 $f(x)\;=\;c_{0}p_{0}(x)+c_{1}p_{1}(x)+c_{2}p_{2}(x)+\ldots$

where the coefficients $c_{n}$ have the expression

 $c_{n}\;=\;\int_{a}^{b}\!f(x)p_{n}(x)W(x)\,dx.$

Other properties

[Not ready . . .]

 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