Hyperbolic wavelets

Abstract

In the last two decades a number of different types of wavelets transforms have been introduced in various areas of mathematics, natural sciences, and technology. These transforms can be generated by means of a uniform principle based on the machinery of harmonic analysis. In this way we pass from the affine group to the wavelet transforms, from the Heisenberg group to the Gábor transform. Taking the congruences of the hyperbolic geometry and using the same method we introduced the concept of hyperbolic wavelet transforms (HWT). These congruences can be expressed by Blaschke functions, which play an eminent role not only in complex analysis but also in control theory. Therefore we hope that the HWT will become an adequate tool in signal and system theories. In this paper we give an overview on some results and applications concerning HWT.