Enrico Casadio Tarabusi

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 ...

Range of the k-Dimensional Radon Transform in Real Hyperbolic Spaces

Abstract : Characterizations of the range of the totally geodesic k-dimensional Radon transform on the n-dimensional hyperbolic space are given both in terms of moment conditions and as the kernel of a differential operator.

Radon transform on spaces of constant curvature

A correspondence among the totally geodesic Radon transforms— as well as among their duals—on the constant curvature spaces is established, and is used here to obtain various range characterizations.

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.

Twist points of planar domains

We establish a potential theoretic approach to the study of twist points in the boundary of simply connected planar domains.

Construction of envelopes of holomorphy for some classes of special domains

The envelope of holomorphy E(Ω) of any domain Ω in Cn which is either of circular type or a tube, is constructed in terms of the envelope of a lower dimensional complex space. As consequences, conditions for the univalence of E(Ω) are proved.

THE HOROCYCLIC RADON TRANSFORM ON NON-HOMOGENEOUS TREES*

This paper studies horocyles on trees and the corresponding Radon transformation. It is seen that a function can be reconstructed from the induced values on the horocycles. A formula is produced for the adjoint transformation and for the inverse.

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.  2003 Elsevier B.V. All rights reserved. MSC: primary 44A12; secondary 05C05, 43A85

Characterization of the range of the Radon transform on homogeneous trees

This article contains results on the range of the Radon transform R on the set H of horocycles of a homogeneous tree T . Functions of compact support on H that satisfy two explicit Radon conditions constitute the image under R of functions of finite support on T . Replacing functions on H by distributions, we extend these results to the non-compact case by adding decay criteria.

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.