What use is $L^2$-convergence for Fourier series?

Question

I'm working through some notes for my signal processing class and there's something elementary that baffles me. We spent lots of hours and dozens of pages setting up the entire theory of Hilbert spaces in order to define the Fourier series of a square integrable periodic function in terms of the orthogonal basis of exponentials $ e_n(t):[0,2\pi] \to \mathbb{C}: t \to e^{int} $.

Then all of the sudden, out of the blue, the notes assault me with a seemingly unrelated theorem about pointwise convergence of the series, and I find out that pointwise convergence isn't guaranteed by $L^2$-convergence. So my (probably naive) question is: what was all that work good for? What use is $L^2$-convergence if it doesn't guarantee pointwise convergence?

Answer 1

A good question! First, even though $L^2$ convergence does not imply pointwise convergence, Parseval's theorem says that the Fourier series converges in that sense, whether or not pointwise. The standard elementary curriculum does seem to urge us to think that the only reasonable notion of convergence of a sequence or infinite sum of functions is pointwise, but that is toooo naive/optimistic.

So, one point is that various manipulations of Fourier series do indeed give correct results with or without pointwise convergence.

"Not all is lost", also, because "Sobolev space theory" is an $L^2$-based extension of the basic $L^2$ theory which does capture pointwise convergence, if one wants.

But/and a very useful technical virtue of $L^2$ is that it is a Hilbert space, in which a (true) "Dirichlet/minimum" principle holds (as opposed to Banach spaces, such as $C^o[0,1]$, which might seem more natural, but in which the Dirichlet/minimum principle fails violently...) Thus, the $L^2$-Sobolev theory can recover pointwise features (continuity, for example), within a Hilbert space context. A good thing.

But/and, again, yes, it was a shock to me years ago when it finally got through to me that $L^2$ convergence and pointwise convergence (e.g., of Fourier series) were not literally/immediately/provably the same thing. Crazy! Who knew? :)

EDIT: while we're here, as @leslietownes rightly comments, there are results about getting pointwise convergence from $L^2$, namely, Lennart Carleson's result that the Fourier series of an $L^2$ function does converge to it almost everywhere. This is a very difficult theorem.

And, while we're here, Kolmogorov constructed a rather tricky $L^1$ function whose Fourier series diverges everywhere. His example is so crazy that it is not possible to succinctly describe it, and I can't even remember the key mechanism. Something number theoretic, in the vein of Diophantine approximation, if I remember at all correctly... Fun stuff, but not easy.

Answer 2

In addition to paul's answer, one thing to keep in mind is that Hilbert spaces are typically meaningless when it comes to function values at individual points. An often glanced over fact is that if you take a function $f\in L^2$ and modify it at a single point $x_0$ calling the result $g$, then $\|f-g\|_2=0$. More generally, the elements of $L^2$ are representatives of the equivalence relation $f\equiv g$ defined by $\|f-g\|_2=0$. This can also be expressed in terms of measure theory, in that $\|f-g\|_2=0$ iff $f$ and $g$ differ on a set of measure 0. So, when speaking about convergence in these spaces, typically the best you can do is almost-everywhere convergence, and not point-wise.