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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.

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3 Answers

up vote 5 down vote accepted

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.

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Does your observation apply fairly equally when it comes to solutions involving the various famous eigenfunction bases - e.g. Legendre, Bessel? On another note, one thing that puzzles me is that, while books do of course give stronger results for particular subtypes of function, they are equally happy to leave the $L^2$ results in their generality. –  Josef K. Apr 30 '11 at 0:40
    
@Josef: In general, one wants to have series approximations whose basis functions have an orthogonality relationship (as is the case for trigonometric functions, Bessel functions, and orthogonal polynomials), since for nice inputs, they behave nicely. –  J. M. Apr 30 '11 at 4:06
    
@J.M: Sorry, I've edited this so you may have seen two versions. When you talk about nice inputs, are the stronger results for particular inputs analogous across different types of Fourier series? For instance, for the trigonometric series we have that if $f$ is twice differentiable and $f''$ is integrable then the convergence is uniform. Does this result apply regardless of orthonormal basis? –  Josef K. Apr 30 '11 at 11:14
    
@Josef: Something like that... of course there may be more intricate conditions, e.g. for a Laguerre expansion, your $f$ should not be "growing" faster than $\exp(x)$... –  J. M. Apr 30 '11 at 11:37
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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.

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The Wikipedia page you linked to leaves a little bit to be desired: for example, I am well aware of the maximal regularity theorems but these have certain assumptions (which are optimal) and they are not detailed in any way on that page. Perhaps zou could supplement this answer with an actual statement of a model result which demonstrates the verity of what you are saying? –  Glen Wheeler Apr 30 '11 at 13:03
    
I found out (superficially) a further advantage to $L^2$ convergence. Sometimes - in circumstances when a solution isn't guaranteed - the series solution may not actually be differentiable. If we know that physically a solution must exist then we can postulate that the solution is imprecise due to the mathematical description of the constraints being somewhat unrealistic. It turns out that the solution based on approximate constraints is a good $L^2$ approximation to the solution of the problem based on the genuine constraints. (summarised from Gonzalez-Velasco, Fourier Analysis and BVPs. –  Josef K. May 1 '11 at 1:28
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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.

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