Clifford Analysis and Integral Geometry

Abstract

In this paper we present a survey of our results in [22] – [27] concerning plane wave decompositions of monogenic functions, leading to Clifford-Radon transforms and including boundary value representations of the classical Radon transform. We also consider integrals of monogenic plane waves depending on parameters (ṯ, $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}\to {s}$) belonging to the product of spheres S ^p−1 × S ^q−1 and prove new plane wave decompositions for the Cauchy kernel.

Finally we replace, monogenic plane waves by differential forms in the parameter ṯ $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}\to {t} + i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}\to {t} \varepsilon N,N$ ∈ N, N the null cone in C^m, leading to a more invariant homo logical treatment of plane wave decompositions. This approach is inspired by ideas from integral geometry in the sense of I.M. Gelfand.