Why convergence of the Fourier Series requires proof, and how it is proven?

Question

The acquisition of Fourier coefficients is better known than proofs of convergence. You set $f(x)$ equal to its Fourier Series and multiplying by a trigonometric function of given frequency, and then integration on both sides will yield the coefficients. This proof in itself is considered not to prove anything about convergence. Not even pointwise convergence?

It was much later that Dirichlet came up with a proof for continuous functions of bounded variety (the function can be split into monotonic subintervals). This proof involves rewriting the Fourier Series to its Dirichlet-kernel form, and the Riemann-Lebesgue Lemma is applied to prove pointwise convergence.

A proof of the pointwise convergence of square integrable functions was also published by Carleson in 1966.

The uniform convergence is generally proven by the Weierstrass M-Test with the help of the Parseval Equality. The criterium is the function shall be twice continuously differentiable (while confusingly other sources say it is enough ig it is once continuously differentiable, can not tell the reason why).

I also heard of proving uniform convergence with the Fejér-kernel and Césaro means.

Is the list complete and true, am I missing something?

Also, some people argue the fact that you can obtain sensible coefficient results in itself should intuitively mean the series should converge. But what interests me is the reasoning why the algebraic proof of the coefficients proves nothing about convergence and why we need these separate proofs.

Answer 1

A few comments are in order:

• You can obtain pointwise convergence of the Fourier Series using the fact that the Fourier coefficients tend to $0$, at least for functions that are differentiable from the left and the right at a point. A simple proof is due to Paul Chernoff. Here is a discussion of that technique: https://www.math.ntnu.no/emner/TMA4120/2008h/felles/FrkConv.pdf
• Dirichlet's proof using the so-called Dirichlet kernel was actually due to Fourier, and it was in his manuscript that was banned from publication for quite some time. Dirichlet probably had access to Fourier's unpublished manuscript; Dirichlet was a student of Fourier.
• Pointwise almost every convergence of the Fourier series to an $L^2$ function is way off the chart in terms of difficulty and depth.