Almost everywhere convergence of a wavelet-type Malmquist-Takenaka series
Gevorg Mnatsakanyan
TL;DR
The paper investigates almost everywhere convergence of wavelet-type Malmquist-Takenaka (MT) series by analyzing the maximal partial sum operator for carefully chosen sequences in the unit disk. The core result proves an $L^2$-boundedness bound for the maximal operator $T^{\mathbf{a_r}}$ with $a_n = r e^{2\pi i n(1-r)}$, uniformly in $r$, which implies a.e. convergence for $f\in L^2(\mathbb{T})$, and extends to a concatenated lacunary sequence $\mathbf{b}$. The proof uses a TT$^*$ argument, linearizes Carleson-type operators via $T_N$, and employs a network of auxiliary symmetries to control a central quadratic form, with a model case guiding the construction. A corollary establishes a.e. convergence for the lacunary MT system, and the work highlights a conformal invariance and the interpolation between Carleson-type and singular-integral regimes; a counterexample shows that certain logarithmic gaps obstruct convergence for other sequences $\mathbf{d_r}$.
Abstract
The Malmquist-Takenaka (MT) system is a complete orthonormal system in $H^2(\mathbf{T})$ generated by an arbitrary sequence of points $a_n$ in the unit disk with $\sum_n (1-|a_n|) = \infty$. The point $a_n$ is responsible for multiplying the $n$th and subsequent terms of the system by a Möbius transform taking $a_n$ to $0$. One can recover the classical trigonometric system, its perturbations or conformal transformations, as particular examples of the MT system. However, many interesting choices of the sequence $a_n$, the MT system is less understood. In this paper, we consider a wavelet-type MT system and prove its almost everywhere convergence in $H^2(\mathbf{T})$.