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In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative "weight function" w precisely if
- <math>\int_{-\infty}^\infty f(x)g(x)w(x)\,dx=0.<math>
In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as
- <math>\langle f,g \rangle=\int_{-\infty}^\infty f(x)g(x)\,w(x)\,dx<math>
then the orthogonal polynomials are simply orthogonal vectors in this inner product space.
A polynomial sequence pn(x) for n = 0, 1, 2, ... , where pn(x) has degree n, is said to be a sequence of orthogonal polynomials with respect to a "weight function" w when any two of them are orthogonal with respect to that weight function, i.e.,
- <math>\langle p_n, p_m \rangle=\int_{-\infty}^\infty p_n(x) p_m(x)\,w(x)\,dx=0\ \mbox{whenever}\ n\neq m.<math>
For example:
- <math>w(x)=1/\sqrt{1-x^2}\ \mbox{for}\ -1
- The Legendre polynomials are orthogonal with respect to the uniform probability distribution on the interval [−1, 1].
See also generalized Fourier series.
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