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

TL;DR

This paper develops a function theory in the real 1+1 dimensional space linked to the principal series representation of SL(2,R), providing analogues of classical complex analysis tools.

Contribution

It introduces a new function theory associated with the principal series of SL(2,R), including counterparts to key complex analysis concepts.

Findings

Constructed a Cauchy integral formula analogue.

Developed Hardy space and Taylor expansion in this setting.

Formulated Cauchy-Riemann equations for the principal series context.

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.