, a polynomial sequence pn
) for n
= 0, 1, 2, ... is said to be a sequence of orthogonal polynomials
with respect to a "weight function" w
In other words, if polynomials are treated as vectors and the inner product
of two polynomials p
) and q
) 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).
- The Legendre polynomials are orthogonal with respect to the uniform probability distribution on the interval [−1, 1].
See also generalized Fourier series