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:
- The Legendre polynomials are orthogonal with respect to the uniform probability distribution on the interval [−1, 1].
See also
generalized Fourier series.