The Algebras Generated by the Laplace Operators in a Semi-homogeneous Tree

In a semi-homogeneous tree, the set of edges is a transitive homogeneous space of the group of automorphisms, but the set of vertices is not (unless the tree is homogeneous): in fact, the latter splits into two disjoint homogeneous spaces V+, V− according to the homogeneity degree. With the goal of constructing maximal abelian convolution algebras, we consider two different algebras of radial functions on semi-homogeneous trees. The first consists of functions on the vertices of the tree: in this case the group of automorphisms gives rise to a convolution product only on V+ and V− separately, and we show that the functions on V+, V− that are radial with respect to the natural distance form maximal abelian algebras, generated by the respective Laplace operators. The second algebra consists of functions on the edges of the tree: in this case, by choosing a reference edge, we show that no algebra that contains an element supported on the disc of radius one is radial, not even in a generalized sense that takes orientation into account. In particular, the two Laplace operators on the edges of a semi-homogeneous (non-homogeneous) tree do not generate a radial algebra, and neither does any weighted combination of them. It is also worth observing that the convolution for functions on edges has some unexpected properties: for instance, it does not preserve the parity of the distance, and the two Laplace operators never commute, not even on homogeneous trees

On the Radon and Riesz transforms in real hyperbolic spaces

Contemporary Mathematics, vol. 14

Integral Geometry in Hyperbolic Spaces and Electrical Impedance Tomography

We study the relation between convolution operators and the totally geodesic Radon transform on hyperbolic spaces. as an application we show that the linearized inverse conductivity problem in the disk can be interpreted exactly in terms of the X- ray transform with respect to the Poincare metric and of a simple convolution operator

The Inverse Conductivity Problem and the Hyperbolic X-Ray Transform

It is shown here how the approximate inversion algorithm of Barber & Brown for the linearized inverse conductivity problem in the disk can be interpreted exactly in terms of the X-ray transform with respect to the Poincare metric and of suitable convolution operators

The circle transform on trees

We consider the overdetermined problem of integral geometry on trees given by the transform that integrates functions on a tree over circles, and exhibit difference equations that describe the range. We then show how this problem modifies if we restrict the transform to some natural subcomplex of the complex of circles, proving inversion formulas and characterizing ranges. (C) 2003 Elsevier B.V. All rights reserved

Representation Theory and Complex Analysis

A collection of advanced articles in Complex Analysis, Lie Groups, Unitary Representations and Quantum Computing, wirtten by the scientific leaders in these areas

Representation Theory and Complex Analysis

A collection of advanced articles in Complex Analysis, Lie Groups, Unitary Representations and Quantum Computing, wirtten by the scientific leaders in these areas