On almost everywhere convergence of Malmquist-Takenaka series

TL;DR

This paper investigates the convergence properties of Malmquist-Takenaka series, a perturbation of classical Fourier series involving M"obius transforms, establishing $L^p$ bounds and connecting to the polynomial Carleson theorem.

Contribution

It provides new $L^p$ bounds for the maximal partial sum operator of Malmquist-Takenaka series under specific zero conditions, linking it to time-frequency analysis and the polynomial Carleson theorem.

Findings

Established $L^p$ bounds for the maximal partial sum operator.

Connected the problem to the polynomial Carleson theorem.

Provided conditions on zeros of M"obius transforms for convergence.

Abstract

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.