On almost everywhere convergence of Malmquist-Takenaka series

Abstract

The Malmquist-Takenaka system is a perturbation of the classical trigonometric system, where powers of $z$ are replaced by products of other Möbius 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 Möbius transforms. We locate the problem in the time-frequency setting and, in particular, we connect it to the polynomial Carleson theorem.

Minor changes following the suggestions of the referee. Accepted to the Journal of Functional Analysis

References in corpus (3)

• Maximal polynomial modulations of singular integrals
• On Zeroes of Random Polynomials and Applications to Unwinding
• The (weak-$L^2$) Boundedness of The Quadratic Carleson Operator