Almost everywhere convergence of a wavelet-type Malmquist-Takenaka series

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 M\"obius 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 $H^2(\mathbf{T})$.