For a set of
square-integrable,
pairwise-orthogonal (with respect to some
weight function <math>w(x)</math>) functions <math>\Phi = \{\phi_n:[a,b]\rightarrow\mathbb{C}\}_{n=0}^\infty</math>, the
generalized Fourier series of a
square-integrable function <math>f:[a,b]\rightarrow\mathbb{C}</math> is
- <math>f(x) \sim \sum_{n=0}^\infty c_n\phi_n(x),</math>
where the coefficients are determined by
- <math>c_n = \left[\int_a^b|\phi(x)|^2dx\right]^{-1}\int_a^bf(x)\phi_n(x)w(x)dx.</math>
The relation <math>\sim</math> becomes equality if <math>\Phi</math> is a complete set[?].
Some theorems on the coefficients <math>c_n</math> include:
- <math>\sum_{n=0}^\infty |c_n|^2\leq\int_a^b|f(x)|^2dx.</math>
If <math>\Phi</math> is a
complete set[?],
- <math>\sum_{n=0}^\infty |c_n|^2 = \int_a^b|f(x)|^2 dx.</math>
See also: complete set[?], orthogonal, square-integrable.