On almost everywhere convergence of Malmquist-Takenaka series

The Malmquist-Takenaka system is a perturbation of the classical trigonometric system, where powers of $z$ are replaced by products of other M"obius transforms of the disc. The system is also inherently connected to the so-called nonlinear phase unwinding decomposition which has been in the center of some recent activity. We prove $L^p$ bounds for the maximal partial sum operator of the Malmquist-Takenaka series under additional assumptions on the zeros of the M"obius transforms. We locate the problem in the time-frequency setting and, in particular, we connect it to the polynomial Carleson theorem.