Curved Radon transforms and factorization of the Veronese equations in Clifford analysis

Abstract

This paper is an updated version of our former article (Sommen, F., 1998, Curved Radon transforms in Clifford analysis. In: Clifford Algebras and their Applications in Mathematical Physics, Fund. Theories Phys., Vol. 94 (Dordrecht: Kluwer Academic Publishers), pp. 369–381). First of all, we study weighted integrals of functions over general surfaces of higher codimension whereby the weights take values in a Clifford algebra. We also introduce and study multi-linear Grassmann and Clifford algebras and apply them to the multi-linear Radon transform. In cases of symmetric multi-linear functions (homogeneous polynomials), we obtain a Clifford analysis generalization of the generalized Radon transform investigated by V.P. Palamodov (Palamodov, V.P., 1994, Radon transformation on real algebraic varieties. In: Gindikin, S. and Michor, P. (Eds) 75 Years of Radon Transform, Lecture Notes in Mathematical Physics, Vol. IV (Boston: International Press), pp. 252–262).

†Dedicated to Richard Delanghe on the occasion of his 65th birthday.

Keywords:

• Radon transform
• Veronese equations
• Grassmann and Clifford algebras

Keywords:

• Mathematics Subject Classifications 2000: 44A12
• 30G35

Notes

†Dedicated to Richard Delanghe on the occasion of his 65th birthday.