Orthogonal polynomials

In mathematics, a polynomial sequence p_n(x) for n = 0, 1, 2, ... is said to be a sequence of orthogonal polynomials with respect to a "weight function" w when

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

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

By convention p_n has degree n; and w should give rise to an inner product, being non-negative and not 0 (see orthogonal).

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

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