Narrow Results

We extend the Funk–Radon–Helgason inversion method of mean value operators to the Radon transform of continuous and L^p functions which are integrated over matrix planes in the space of real rectangular matrices. Necessary and sufficient conditions of existence of for such f and explicit inversion formulas are obtained. New higher-rank phenomena related to this setting are investigated.

We study when the twisted groupoid Banach *-algebra $L^1({𝒢},\sigma )$ is Hermitian. In particular, we prove that Hermitian groupoids satisfy the weak containment property. Furthermore, we find that for $L^1({𝒢},\sigma )$ to be Hermitian it is sufficient that $L^1({{𝒢}}_{\sigma })$ is Hermitian. Moreover, if ${𝒢}$ is ample, we find necessary conditions for $L^1({𝒢},\sigma )$ to be Hermitian in terms of the fibers ${{𝒢}}_x^x$.

This paper proves the existence of L^p solution to the fuzzy functional equation

A new generalization of the Ostrowski–Gruss inequality is introduced in three different cases for functions in L¹[a, b] and L^∞[a, b] spaces and its application is given for deriving error bounds of some quadrature rules.

Approximation theory constitutes a useful field that is related to quasi all other fields, in both theoretical and applied sciences. In approximation theory, the aim is generally to construct an idea about a function that is usually impossible or difficult to evaluate directly, and which is usually unknown. Such functions appear widely in PDE, probability law distributions, statistical modeling, etc. Some of the most known approximators nowadays are neural networks and wavelets, which constitute good classes of elementary functions permitting as efficiently as possible to describe functions in appropriate spaces. This paper aims to develop combined neural networks and wavelet approximators for functions, based on the involvement of wavelets as activation functions. Some necessary conditions on the activation function to approximate $L^p(\mu )$ and $W^{m,p}(\mu )$-elements are relaxed as well as those on the measure $\mu$. We prove that for a wavelet activation function, any element of $L^p(\mu )$ as well as $W^{m,p}(\mu )$ can be well approximated for arbitrary measures $\mu$. The theoretical results are subject to an experimental application in order to show their effectiveness.

A locally compact group $G$ has property PL if every isometric $G$-action either has bounded orbits or is (metrically) proper. For $p\ge 1$, say that $G$ has property BP_p if the same alternative holds for the smaller class of affine isometric actions on $L^p$-spaces. We explore properties PL and BP_p and prove that they are equivalent for some interesting classes of groups: abelian groups, amenable almost connected Lie groups, amenable linear algebraic groups over a local field of characteristic 0. The appendix provides new examples of groups with property PL, including nonlinear ones.

We describe the main interpolation methods for Banach couples, their roots and some of their properties and applications. We consider also factorization of weakly compact operators and interpolation of couples of Banach algebras.