Editing Generalized Fourier series (section)

== Sturm-Liouville Problems ==
Given the space <math> L^2(a,b) </math> of square integrable functions defined on a given interval, one can find orthogonal bases by considering a class of boundary value problems on the interval <math> [a,b] </math> called [[Sturm-Liouville problems|regular Sturm-Liouville problems]]. These are defined as follows, 
<math display="block">
(rf')' + pf + \lambda wf = 0
</math>
<math display="block">
B_1(f) = B_2(f) = 0
</math>
where <math> r, r'</math> and <math> p </math> are real and continuous on <math> [a,b] </math> and <math> r > 0 </math> on <math> [a,b] </math>, <math> B_1 </math> and <math> B_2 </math> are [[self-adjoint]] boundary conditions, and <math> w </math> is a positive continuous functions on <math> [a,b] </math>.

Given a regular Sturm-Liouville problem as defined above, the set <math> \{\phi_n\}_{1}^{\infty} </math> of [[eigenfunctions]] corresponding to the distinct [[eigenvalue]] solutions to the problem form an orthogonal basis for <math> L^2(a,b) </math> with respect to the weighted inner product <math> \langle\cdot,\cdot\rangle_w </math>.<ref>Folland p.89</ref> We also have that for a function <math> f \in L^2(a,b) </math> that satisfies the boundary conditions of this Sturm-Liouville problem, the series <math> \sum_{n=1}^{\infty} \langle f,\phi_n \rangle \phi_n </math> [[converges uniformly]] to <math> f </math>.<ref>Folland p.90</ref>