[For the entire collection see Zbl 0742.00067.]Let $P^m$ be the set of hyperplanes $\sigma :\langle \overrightarrow{x},\overrightarrow{\theta }\rangle =p$ in ${ℛ}^m$, $S^{m-1}$ the unit sphere of ${ℛ}^m$, $E^m$ the exterior of the unit ball, $T^m$ the set of hyperplanes not passing through the unit ball, $Rf(\overrightarrow{\theta },p)={\int }_{\sigma }f\left(\overrightarrow{x}\right)d\overrightarrow{x}$ the Radon transform, $R^{\#}g\left(\overrightarrow{x}\right)={\int }_{S^{m-1}}g(\overrightarrow{\theta },\langle \overrightarrow{x},\overrightarrow{\theta }\rangle )d{S}_{\overrightarrow{\theta }}$ its dual. $R$ as operator from $L^2\left({ℛ}^m\right)$ to $L^2(S^{m-1)}\times ℛ)$ is a closable, densely defined operator, $R^{*}$ denotes the operator given by $\left(R^{*}g\right)\left(\overrightarrow{x}\right)=R^{\#}g\left(\overrightarrow{x}\right)$ if the integral exists for $\overrightarrow{x}\in {ℛ}^m$ a.e. Then the closure of $R^{*}$ is the adjoint of $R$. The author shows that the Radon transform and its dual can be linked by two operators...

The profinite topology on any abstract group $G$, is one such that the fundamental system of neighborhoods of the identity is given by all its subgroups of finite index. We say that a group $G$ has the Ribes-Zalesskii property of rank $k$, or is RZ${}_k$ with $k$ a natural number, if any product $H_1{H}_2\cdots H_k$ of finitely generated subgroups $H_1,H_2,\cdots ,H_k$ is closed in the profinite topology on $G$. And a group is said to have the Ribes-Zalesskii property or is RZ if it is RZ${}_k$ for any natural number $k$. In this paper we characterize groups...

We find the minimal real number k such that the kth power of the Fourier transform of any continuous, orbital measure on a classical, compact Lie group belongs to l2. This results from an investigation of the pointwise behaviour of characters on these groups. An application is given to the study of Lp-improving measures.

Pointwise upper bounds for characters of compact, connected, simple Lie groups are obtained which enable one to prove that if μ is any central, continuous measure and n exceeds half the dimension of the Lie group, then ${\mu }^n\in L^1$. When μ is a continuous, orbital measure then ${\mu }^n$ is seen to belong to $L^2$. Lower bounds on the p-norms of characters are also obtained, and are used to show that, as in the abelian case, m-fold products of Sidon sets are not p-Sidon if p < 2m/(m+1).