Analysis in $R^{1,1}$ or the Principal Function Theory

Abstract

We explore a function theory connected with the principal series representation of SL(2,R) in contrast to standard complex analysis connected with the discrete series. We construct counterparts for the Cauchy integral formula, the Hardy space, the Cauchy-Riemann equation and the Taylor expansion. Keywords: Complex analysis, Cauchy integral formula, Hardy space, Taylor expansion, Cauchy-Riemann equations, Dirac operator, group representations, SL(2,R), discrete series, principal series, wavelet transform, coherent states.

25 pages, 2 figures. 14/12/97: Language corrections

Cited by in corpus (15)

• Erlangen Programme at Large: An Overview
• Erlangen Program at Large-1: Geometry of Invariants
• Meeting Descartes and Klein Somewhere in a Noncommutative Space
• Erlangen Program at Large--2: Inventing a wheel. The parabolic one
• Elliptic, Parabolic and Hyperbolic Analytic Function Theory--0: Geometry of Domains
• Wavelets in Banach Spaces
• Covariant Transform
• Cross-Toeplitz Operators on the Fock--Segal--Bargmann Spaces and Two-Sided Convolutions on the Heisenberg Group
• Spaces of Analytical Functions and Wavelets--Lecture Notes
• Nilpotent Lie Groups in Clifford Analysis: Five Directions for Research
• Calculus of Operators: Covariant Transform and Relative Convolutions
• Erlangen Program at Large: Outline
• Spectrum as the Support of Functional Calculus
• Monogenic Calculus as an Intertwining Operator
• Notions of Regularity for Functions of a Split-Quaternionic Variable