It is proved that every real metrizable locally convex space which is not nuclear contains a closed additive subgroup K such that the quotient group G = (span K)/K admits a non-trivial continuous positive definite function, but no non-trivial continuous character. Consequently, G cannot satisfy any form of the Bochner theorem.

On considère l’espace $E=L^2({\Psi }_2.{\Psi }_1^{-1}dx)$ où ${\Psi }_2$ et ${\Psi }_1$ sont deux fonctions définies-négatives, réelles et continues sur ${\mathbf{R}}^n$. On étudie la possibilité d’approcher, au sens de la norme de $E$, tout élément $\phi$ de $E$ par des combinaisons linéaires d’éléments de $E$ qui sont transformés de Fourier de mesures positives de support inclus dans le spectre de $\phi$. Des méthodes de théorie du potentiel permettent de donner une réponse positive (sous certaines hypothèses additionnelles). On obtient ainsi des généralisations, au cas de ${\mathbf{R}}^n$,...

Let $G={*}_{j=1}^{q+1}{G}_{n_j+1}$ be the product of $q+1$ finite groups each having order $n_j+1$ and let $\mu$ be the probability measure which takes the value $p_j/n_j$ on each element of $G_{n_j+1}\setminus \left\{e\right\}$. In this paper we shall describe the point spectrum of $\mu$ in $C_{\mathrm{reg}}^{*}\left(G\right)$ and the corresponding eigenspaces. In particular we shall see that the point spectrum occurs only for suitable choices of the numbers $n_j$. We also compute the continuous spectrum of $\mu$ in $C_{\mathrm{reg}}^{*}\left(G\right)$ in several cases. A family of irreducible representations of $G$, parametrized on the continuous spectrum of $\mu$,...

The affine group of a local field acts on the tree $𝕋\left(𝔉\right)$ (the Bruhat-Tits building of $\mathrm{GL}(2,𝔉)$) with a fixed point in the space of ends $\partial 𝕋\left(F\right)$. More generally, we define the affine group $Aff\left(𝔉\right)$ of any homogeneous tree $𝕋$ as the group of all automorphisms of $𝕋$ with a common fixed point in $\partial 𝕋$, and establish main asymptotic properties of random products in $Aff\left(𝔉\right)$: (1) law of large numbers and central limit theorem; (2) convergence to $\partial 𝕋$ and solvability of the Dirichlet problem at infinity; (3) identification of the Poisson boundary...

We investigate random Sidon-type sets in which the degrees of the representations are weighted. These variants of Sidon sets are of interest as there are compact non-abelian groups which admit no infinite Sidon sets. In this note we determine the largest weight function such that infinite random weighted Sidon sets exist in all infinite compact groups.

In this paper we study the Radon transform $R$ on the set $\mathscr{H}$ of horocycles of a homogeneous tree $T$, and describe its image on various function spaces. We show that the functions of compact support on $\mathscr{H}$ that satisfy two explicit Radon conditions constitute the image under $R$ of functions of finite support on $T$. We extend these results to spaces of functions with suitable decay on $T$, whose image under $R$ satisfies corresponding decay conditions and contains distributions on $\mathscr{H}$ that are not defined pointwise....