Hilbert's trip to the Casino

To mathematicians, the Hilbert transform is a singular integral operator that gives access to harmonic conjugate functions via a convolution of boundary values. To others, it may mean a signal transformer that produces an audible signal, sent to a radio's speaker or, it might model the trajectories of charged particles in a potential. The concept of martingales on the other hand, is often associated with old betting strategies in so-called fair games.

In part of this lecture we focus on connections between the Hilbert transform and transforms on systems of martingales.

We give an idea of various results that were obtained using this connection between analysis and probability. Applications include characterisations of certain Banach spaces, norm estimates in high dimensions and so called weighted estimates with applications to PDE and multivariate stationary stochastic processes.