Editing Generalized Fourier series (section)

== Definition ==

Consider a set <math>\Phi = \{\phi_n:[a,b] \to \mathbb{C}\}_{n=0}^\infty</math> of [[square-integrable]] complex valued functions defined on the closed interval <math> [a,b] </math> that are pairwise [[orthogonal]] under the weighted [[inner product]]:

<math>\langle f, g \rangle_w = \int_a^b f(x) \overline{g(x)} w(x) dx,</math>

where <math>w(x)</math> is a [[weight function]] and <math>\overline g</math> is the [[complex conjugate]] of <math> g </math>. Then, the '''generalized Fourier series''' of a function <math> f </math> is:
<math display="block">f(x) = \sum_{n=0}^\infty c_n\phi_n(x),</math>where the coefficients are given by:
<math display="block">c_n = {\langle f, \phi_n \rangle_w\over \|\phi_n\|_w^2}.</math>