2002/04/01 by Vladimir V. Kisil, Kisil, Vladimir V.
Mathematics · Physics and Astronomy · #30G30 #42C40 #43A85 #46H30 #47A13 #81R30 #81R60 #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #math-ph #math.CV #math.FA #math.MP #msc:30G30 #msc:42C40 #msc:43A85 #msc:46H30 #msc:47A13 #msc:81R30 #msc:81R60
paper · pdf · doi:10.48550/arxiv.math/0204018
LaTeX, pages 92, two PS picture
arxiv created 2002/04/01 · openalex publication_date 2002/04/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Segal-Bargmann space). We also briefly discuss applications to the operator theory (functional calculus) and quantum mechanics.