We study natural measures on sets of $\beta$-expansions and on slices through self similar sets. In the setting of $\beta$-expansions, these allow us to better understand the measure of maximal entropy for the random $\beta$-transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading...

The inverse Fast Fourier Transform is a common procedure to solve a convolution equation provided the transfer function has no zeros on the unit circle. In our paper we generalize this method to the case of a singular convolution equation and prove that if the transfer function is a trigonometric polynomial with simple zeros on the unit circle, then this method can be extended.

The paper is concerned with the spectral analysis for the class of linear operators $A=D_{\lambda }+X\otimes Y$ in non-archimedean Hilbert space, where $D_{\lambda }$ is a diagonal operator and $X\otimes Y$ is a rank one operator. The results of this paper turn out to be a generalization of those results obtained by Diarra.

We study the relation between standard ideals of the convolution Sobolev algebra ${₊}^{\left(n\right)}\left(tⁿ\right)$ and the convolution Beurling algebra L¹((1+t)ⁿ) on the half-line (0,∞). In particular it is proved that all closed ideals in ${₊}^{\left(n\right)}\left(tⁿ\right)$ with compact and countable hull are standard.

Considérons un noyau de convolution $\kappa \ne 0$ sur ${\mathbf{R}}^n$ ($n\ge 3$) sphériquement symétrique et vérifiant $\Delta \kappa \ge 0$ en dehors de 0, qui s’appelle un noyau de Frostman-Kunugui. Le but de cet article est de donner les conditions suffisantes pour le principe du balayage de $\kappa$.Supposons que $\kappa$ est de classe $C^2$ en dehors de 0 et s’annule à l’infini. Si $\Delta \kappa$ vérifie la conditions suivante (*), alors $\kappa ={\kappa }_0+c{r}^{2-n}$, où ${\kappa }_0$ est un noyau de Dirichlet et où $c$ est une constante $\ge 0$.(*) $\Delta \kappa =0$ en dehors de 0 ou bien $\Delta \kappa \>0$ en dehors de 0 et, pour $t\>0$ quelconque, $\frac{\Delta \kappa *s_t\left(x\right)}{\Delta \kappa \left(x\right)}$ décroît...

Soit $\mu \in M\left(G\right)$, algèbre de convolution des mesures de Radon bornées sur le groupe abélien localement compact $G$. Pour que $\mu *M\left(G\right)$ soit fermé dans $M\left(G\right)$ (ou, ce qui revient au même, pour que $\mu *L^1\left(G\right)$ soit fermé), il faut et il suffit que $\mu$ soit la convolution d’une mesure inversible et d’une mesure idempotente.