# Fourier Series Theorem

I had many things that I wanted to talk about, but when I read this theorem, it was so shocking that I just had to post it. Now from a general intuition standpoint, you might think this theorem to be quite natural. But remember, most of us have been trained to think Fourier series are extremely nicely behaved. In fact, if I had read the wikipedia article when I was learning Fourier series years ago, I wouldn’t be surprised as much since under Divergence, it tells us this stuff.

So secretly we always are working with $L^2(T)$, or square-integrable $2\pi$-periodic functions. When we think in this way, we have the Fourier series of f at x given by the partial sums $s_n(f; x)=\frac{1}{2\pi}\int_{-\pi}^\pi f(t)D_n(x-t)dt$ where $D_n(t)=\sum_{k=-n}^n e^{ikt}$. It turns out quite simply that in $L^2$-norm the partial sums converge to f quite quickly. This nice convergence tricks us into thinking that we will have nice convergence all the time.

So if we switch to just continuous $2\pi$-periodic functions (which is a dense subset of $L^2(T)$), do we get something as simple as point-wise convergence (it would surely be too much to ask for uniform convergence)? Well, from a common measure theory theorem, since the sequence converges in $L^2$-norm we have a subsequence converging point-wise almost everywhere. But this leaves much room for error. How much error you ask?

It turns out that there is is a set $E\subset C(T)$ which is a dense $G_\delta$ in C(T) which has the following property: For each $f\in E$, the set $Q_f=\{x: s^*(f; x)=\infty\}$ is a dense $G_\delta$ in $\mathbb{R}$. Note that $s^*(f;x)=sup_n |s_n(f;x)|$.

This is pretty rough considering it means in non-technical terms that continuous functions are completely filled with functions for which the points where the Fourier series behaves badly is almost everywhere. Also, note that a “dense $G_\delta$” is uncountable (nice little topological proof if you want to try it), so this isn’t like some minimally dense set we’re talking about.

I was going to prove the theorem, but now I don’t think I will because, there may be at best one person that has made it this far. If interested drop a comment and I’ll gladly add the proof here, though. I just need to make sure there is an interest before going forth with it.

We should note that it doesn’t take much to correct the problems stated here. If we just make sure that our function is Lipschitz of some order, then we have a convergent Fourier series.