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.
Referenced By
List of mathematical topics (M-O) | List of polynomial topics | List of real analysis topics
|
