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.