The existence of an $L^2$ orthonormal basis consisting of eigenfunctions of a Sturm-Liouville equation helps us to express the solutions of various ODEs and PDEs as infinite series. However, in the general case these infinite series converge to the said solution only in the sense of mean square convergence.
Why is it useful to have series representations of solutions which converge only 'in the mean'? I can think of a few reasons:
- because sometimes it may be the best we can do;
- because it's a stepping stone to proving stronger types of convergence for particular cases;
- because we can be happy that they are 'generally' correct, although they can be atrociously wrong particularly at single (zero-measure) points.
However, I can't help noticing that the existence of $L^2$-convergent series as solutions to equations - and more generally approximations to $L^2$ functions - are celebrated in their own right as a practical, applicable achievement. I would welcome people's thoughts about the direct use of these solutions, especially how the general idea of being 'comfortable with these solutions on average' translates into some kind of practical reliability.