A "good'' periodic function can be represented via its Fourier series

$ f(x) = \sum_{k\in \mathbb{Z}} \hat{f}(k) e^{2\pi i k x} .$

When we hear different tones, this is what our ear does: it hears the Fourier modes.

However, what are the functions for which the above representation actually holds?

Suppose that $f \in L^{2}\bigl([0,1)\bigr)$. It is relatively easy to see that the above equality holds in $L^{2}\bigl([0,1)\bigr)$ sense using basic functional analysis of Hilbert spaces. Once again, functional analysis tells us that the above equality fails to hold pointwise for every $x$.

In 1913 Luzin conjectured that the convergence of the Fourier series does hold for almost every $x\in[0,1)$ (i.e. with exception of a set of vanishing measure). But it was only in 1966 the Lennart Carleson, convinced that the conjecture was false, while constructing a counterexample accidentally proved that the convergence actually holds.

We will talk about how Carleson's insanely complicated proof gave origin to what is known as time-frequency analysis and phase-space portrait analysis, an extremely useful toolbox for dealing with singular and oscillatory integral operators.