Generalized hypercomplex function theory

Abstract:

Lipman Bers and Ilya Vekua extended the concept of an analytic function by considering the distributional solutions of elliptic systems of two equations with two unknowns and two independent variables. These solutions have come to be known as generalized (or pseudo) analytic functions. Subsequently, Avron Douglis introduced an algebra and a class of functions which satisfy (classically) the principal part of an elliptic system of 2r equations with 2r unknowns and two independent variables. In Douglis’ algebra these systems of equations can be represented by a single “hypercomplex” equation. Solutions of such equations are termed hyperanalytic functions. In this work, the class of functions studied by Douglis is extended in a distributional sense much in the same way as Bers and Vekua extended the analytic functions. We refer to this extended class of functions as the class of generalized hyperanalytic functions.