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

The Malmquist-Takenaka (MT) system is a complete orthonormal system in H²(T) generated by an arbitrary sequence of points aₙ in the unit disk with ∑ₙ (1-|aₙ|) = ∞. The point aₙ is responsible for multiplying the nth and subsequent terms of the system by a M\"obius transform taking aₙ 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 aₙ, the MT system is less understood. In this paper, we consider a wavelet-type MT system and prove its almost everywhere convergence in H²(T).