Multiresolution in the Bergman space

TL;DR

This paper introduces a multiresolution framework in Bergman space using a set of poles, enabling practical hyperbolic wavelet representations of signals with convergence analysis.

Contribution

It proposes a novel multiresolution construction in Bergman space based on a set of poles, linking it to wavelet theory and the Blaschke group.

Findings

Provides conditions for sampling and interpolation in Bergman space.

Establishes a discretization scheme analogous to affine wavelets.

Analyzes convergence properties of the hyperbolic wavelet representation.

Abstract

In this paper we give a multiresolution construction in Bergman space. The successful application of rational orthogonal bases needs a priori knowledge of the poles of the transfer function that may cause a drawback of the method. We give a set of poles and using them we will generate a multiresolution in $A^2$. We study the upper and lower density of this set, and we give sufficient conditions for this set to be interpolating or sampling sequence for the Bergman space. The construction is an analogy with the discrete affine wavelets, and in fact is the discretization of the continuous voice transform generated by a representation of the Blaschke group over the Bergman space. The constructed discretization scheme gives opportunity of practical realization of hyperbolic wavelet representation of signals belonging to the Bergman space if we can measure their values on a given set of points inside the unit disc. Convergence properties of the hyperbolic wavelet representation will be studied.