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Orthogonal polynomials

In mathematics, a polynomial sequence pn(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 pn has degree n; and w should give rise to an inner product, being non-negative and not 0 (see orthogonal).

For example:

See also generalized Fourier series.