Abstract
In the complex case, the Blaschke group was introduced and studied. It turned out that in the complex case this group plays important role in the construction of analytic wavelets and multiresolution analysis in different analytic function spaces. The extension of the wavelet theory to quaternion variable function spaces would be very beneficial in the solution of many problems in physics. A first step in this direction is to give the quaternionic analogue of the Blaschke group. In this paper we introduce the quaternionic Blaschke group and we study the properties of this group and its subgroups.
Highlights
We write D for the unit disc D := {z ∈ C : |z| < 1}, and T := {z ∈ C : |z| = 1} for the unit circle.The element (e := (0, 1) ∈ B) will play a special role
In the parameter set B := D × T let us define the operation induced by the function composition in the following way: Ba1 ◦a2 := Ba1 ◦ Ba2
One reason is that the techniques of the complex analysis can be applied more directly in the study of the properties of the voice transforms generated by representations of the Blaschke group on different analytic function spaces
Summary
One reason is that the techniques of the complex analysis can be applied more directly in the study of the properties of the voice transforms (so called hyperbolic wavelet transforms) generated by representations of the Blaschke group on different analytic function spaces (see [7,8,9,10,11]). Obviously Qc (z) = Q∗c (z) (z ∈ C) This implies that the map Qc is an isometric isomorphism between the fields C and Qc. The complex numbers and their extensions, the quaternions are very useful in the description of many problems in geometry and physics.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
