Spaces of Analytical Functions and Wavelets Lecture Notes

This is (raw) lecture notes of the course read on 6th European intensive course on Complex Analysis (Coimbra, Portugal) in 2000. Our purpose is to describe a general framework for generalizations of the complex analysis. As a consequence a classification scheme for different generalizations is obtained. The framework is based on wavelets (coherent states) in Banach spaces generated by “admissible” group representations. Reduced wavelet transform allows naturally describe in abstract term main objects of an analytical function theory: the Cauchy integral formula, the Hardy and Bergman spaces, the Cauchy-Riemann equation, and the Taylor expansion. Among considered examples are classical analytical function theories (one complex variables, several complex variables, Clifford analysis, Segal-Bargmann space) as well as new function theories which were developed within our framework (function theory of hyperbolic type, Clifford version of SegalBargmann space). We also briefly discuss applications to the operator theory (functional calculus) and quantum mechanics. 2000 Mathematics Subject Classification. Primary: 30G30; Secondary: 42C40, 43A85, 46H30, 47A13, 81R30, 81R60.