Hyperbolic Wavelets and Multiresolution in $H^{2}(\mathbb{T})$

Abstract

In signal processing and system identification for $H^{2}(\Bbb{T})$ and $H^{2}(\Bbb{D})$ the traditional trigonometric bases and trigonometric Fourier transform are replaced by the more efficient rational orthogonal bases like the discrete Laguerre, Kautz and Malmquist-Takenaka systems and the associated transforms. These bases are constructed from rational Blaschke functions, which form a group with respect to function composition that is isomorphic to the Blaschke group, respectively to the hyperbolic matrix group. Consequently, the background theory uses tools from non-commutative harmonic analysis over groups and the generalization of Fourier transform uses concepts from the theory of the voice transform. 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. In this paper we give a set of poles and using them we will generate a multiresolution in $H^{2}(\Bbb{T})$ and $H^{2}(\Bbb{D})$. 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 space $H^{2}(\Bbb{T})$. The constructed discretization scheme gives opportunity of practical realization of hyperbolic wavelet representation of signals belonging to $H^{2}(\Bbb{T})$ and $H^{2}(\Bbb{D})$ if we can measure their values on a given set of points inside the unit circle or on the unit circle. Convergence properties of the hyperbolic wavelet representation will be studied.