Orthogonal polynomials

In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative "weight function" w precisely if

$\int_{-\infty}^\infty f(x)g(x)w(x)\,dx=0.$

In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as

$\langle f,g \rangle=\int_{-\infty}^\infty f(x)g(x)\,w(x)\,dx$

then the orthogonal polynomials are simply orthogonal vectors in this inner product space.

A polynomial sequence p_n(x) for n = 0, 1, 2, ... , where p_n(x) has degree n, is said to be a sequence of orthogonal polynomials with respect to a "weight function" w when any two of them are orthogonal with respect to that weight function, i.e.,

$\langle p_n, p_m \rangle=\int_{-\infty}^\infty p_n(x) p_m(x)\,w(x)\,dx=0\ \mbox{whenever}\ n\neq m.$

For example:

The Hermite polynomials are orthogonal with respect to a normal probability distribution.

The Chebyshev polynomials are orthogonal with respect to the weight function

$w(x)=1/\sqrt{1-x^2}\ \mbox{for}\ -1<x<1.$

The Legendre polynomials are orthogonal with respect to the uniform probability distribution on the interval [−1, 1].