Computerized tomograhphy is a technique for computation and visualization of density (i.e. X- or $\gamma$-ray absorption coefficients) distribution over a cross-sectional anatomic plane from a set of projections. Three-dimensional reconstruction may be obtained by using a system of parallel planes. For the reconstruction of the transverse section it is necessary to choose an appropriate method taking into account the geometry of the data collection, the noise in projection data, the amount of data, the...

By using the notion of contraction of Lie groups, we transfer $L^p-L²$ estimates for joint spectral projectors from the unit complex sphere $S^{2n+1}$ in ${ℂ}^{n+1}$ to the reduced Heisenberg group hⁿ. In particular, we deduce some estimates recently obtained by H. Koch and F. Ricci on hⁿ. As a consequence, we prove, in the spirit of Sogge’s work, a discrete restriction theorem for the sub-Laplacian L on hⁿ.

Dans cet article on étudie en premier lieu la résolvante (le noyau de Green) d’un opérateur agissant sur un arbre localement fini. Ce noyau est supposé invariant par un groupe $G$ d’automorphismes de l’arbre. On donne l’expression générique de cette résolvante et on établit des simplifications sous différentes hypothèses sur $G$.En second lieu on introduit la transformation de Poisson qui associe à une mesure additive finie sur l’espace $\Omega$ des bouts de l’arbre une fonction propre de l’ opérateur. On...

Let $Z\left(\mathrm{Ann}\right(r,R\left)\right)$ be the class of all continuous functions $f$ on the annulus $\mathrm{Ann}(r,R)$ in ${ℂ}^n$ with twisted spherical mean $f\times {\mu }_s\left(z\right)=0,$ whenever $z\in {ℂ}^n$ and $s\>0$ satisfy the condition that the sphere $S_s\left(z\right)\subseteq \mathrm{Ann}(r,R)$ and ball $B_r\left(0\right)\subseteq B_s\left(z\right).$ In this paper, we give a characterization for functions in $Z\left(\mathrm{Ann}\right(r,R\left)\right)$ in terms of their spherical harmonic coefficients. We also prove support theorems for the twisted spherical means in ${ℂ}^n$ which improve some of the earlier results.

We give a characterization of the pairs of weights (v,w), with w in the class $A_{\infty }$ of Muckenhoupt, for which the fractional maximal function is a bounded operator from $L^p(X,vd\mu )$ to $L^q(X,wd\mu )$ when 1 < p ≤ q < ∞ and X is a space of homogeneous type.

We study unbounded harmonic functions for a second order differential operator on a homogeneous manifold of negative curvature which is a semidirect product of a nilpotent Lie group N and A = ℝ⁺. We prove that if F is harmonic and satisfies some growth condition then F has an asymptotic expansion as a → 0 with coefficients from 𝓓'(N). Then we single out a set of at most two of these coefficients which determine F. Then using asymptotic expansions we are able to prove some theorems...

Let $X$ be a Riemannian symmetric space of the noncompact type. We prove that the solution of the time-dependent Schrödinger equation on $X$ with square integrable initial condition $f$ is identically zero at all times $t$ whenever $f$ and the solution at a time $t_0\&gt;0$ are simultaneously very rapidly decreasing. The stated condition of rapid decrease is of Beurling type. Conditions respectively of Gelfand-Shilov, Cowling-Price and Hardy type are deduced.

We prove that the first reduced cohomology with values in a mixing $L^p$-representation, $1\&lt;p\&lt;\infty$, vanishes for a class of amenable groups including connected amenable Lie groups. In particular this solves for this class of amenable groups a conjecture of Gromov saying that every finitely generated amenable group has no first reduced ${\ell }^p$-cohomology. As a byproduct, we prove a conjecture by Pansu. Namely, the first reduced $L^p$-cohomology on homogeneous, closed at infinity, Riemannian manifolds vanishes. We also...