Why is it useful to express PDE solutions as $L^2$-convergent series?

Question

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.

Thank you.

Answer 1

Since it's quite rare to get "closed form" solutions, series approximations are often the best you can do. Yes, in general they may not always converge pointwise, but in fact, except for pathological cases, if the inputs are smooth they actually will converge pointwise. For example, you have to work pretty hard to come up with an explicit example of a periodic continuous function whose Fourier series does not converge pointwise.

Answer 2

There are "regularity" theorems about $L^2$ solutions to elliptic PDEs being "smoother" than just $L^2$. For example, one might be able to prove the partial derivatives are $L^2$, and with enough regularity/smoothness one concludes that the function itself is continuous. So the theory of "weak" solutions can be a "stepping stone" to proving a "strong" solution.

Answer 3

In my experience, it seems that the Hilbert-space context (e.g., Plancherel theorems) allows the most decisive assertions, even if they do not respond quite directly to the primordial questions. This slight disconnect is well-known with Fourier series, and with Sturm-Liouville problems.

I remember being shocked, "in my youth", that $L^2$ convergence was not the same as pointwise, or as uniform pointwise. It seemed a hostile act on nature's part.

However, eventually, for example by looking at somewhat modern theory of linear, especially elliptic, PDE, one sees that "$L^2$-differentiation" is entirely tolerable, and, invoking Sobolev's inequalities at moment where necessary, quite reasonable assertions about "classical" differentiability can be cashed-in when necessary.

I learned how simple Sobolev theory can be from G. Folland's Tata notes on PDE. (W. Rudin's "Functional Analysis" seems not to appreciate Sobolev theory.) The simplest possible case, of Fourier series in one variable, is written out in detail sufficient for students who aren't seasoned analysts, in my notes Functions on circles .

Even in more serious structured situations, one can hope for a Plancherel theorem for $L^2$. There is no reasonable general expectation for other $L^p$ spaces, really. Thus, $L^2$ Sobolev theory has some universality not shared by other viewpoints.