Hyperbolic wavelet frames and multiresolution in the weighted Bergman spaces

Abstract

In this paper, we construct so-called hyperbolic wavelet frames in weighted Bergman spaces and a multiresolution analysis (MRA) generated by them. The construction is based on a new example of sampling set for the weighted Bergman space, which is related to the Blaschke group operation. The introduced MRA is an analog of the MRA generated by the affine wavelets in the space of the square integrable functions on the real line, and in fact is the discretization of the continuous voice transform generated by a representation of the Blaschke group over the weighted Bergman space. The projection to the resolution levels is an interpolation operator. This projection operator gives opportunity of practical realization of the hyperbolic wavelet representation of a function belonging to the weighted Bergman space, if we can measure the values of the function on a given set of points inside the unit disc. Convergence properties of the hyperbolic wavelet representation are studied.