Using a description of the topology of the spaces ${\mathbf{E}}^{\text{'}}\left(\Omega \right)$ ($\Omega$ open convex subset of $R^n$) via the Fourier transform, namely their analytically uniform structures, we arrive at a formula describing the convex hull of the singular support of a distribution $T$, $T\in {\mathbf{E}}^{\text{'}}$. We give applications to a class of distributions $T$ satisfying$$\text{cv.}\text{sing.}\text{supp.}S*T=\text{cv.}\text{sing.}\text{supp.}S+\text{cv.}\text{sing.}\text{supp.}T$$for all $S\in {\mathbf{E}}^{\text{'}}$.

Let $F$ and $G$ be distributions in ${𝒟}^{\text{'}}$ and let $f$ be an infinitely differentiable function with $f^{\text{'}}\left(x\right)>0$, (or $<0$). It is proved that if the neutrix product $F\circ G$ exists and equals $H$, then the neutrix product $F\left(f\right)\circ G\left(f\right)$ exists and equals $H\left(f\right)$.

It is shown that every closed rotation and translation invariant subspace $V$ of $C\left({\mathbf{R}}^n\right)$ or $\delta \left({\mathbf{R}}^n\right)$, $n\ge 2$, is of spectral synthesis, i.e. $V$ is spanned by the polynomial-exponential functions it contains. It is a classical problem to find those measures $\mu$ of compact support on ${\mathbf{R}}^2$ with the following property: (P) The only function $f\in C\left({\mathbf{R}}^2\right)$ satisfying ${\int }_{{\mathbf{R}}^2}f\circ \sigma d\mu =0$ for all rigid motions $\sigma$ of ${\mathbf{R}}^2$ is the zero function. As an application of the above result a characterization of such measures is obtained in terms of their Fourier-Laplace transforms....

We construct a solution T0 in the distribution sense of equation fT = 1 near a critical point of f and we give an upper bound for the order of T0 in terms of f's Newton Polyhedron, provided f is non degenerate in some sense. The order of T0 is equal to this upper bound when f is non-negative.

Étant donnés $r$ champs de vecteurs $X_1,...,X_r$, réels, de classe $C^{\infty }$ dans ${\mathbf{R}}^n$, nous étudions l’existence de traces sur une variété de classe $C^{\infty }$, de dimension $(n-1)$, frontière d’un ouvert $\Omega$, des distributions $u\in {𝒟}^{\text{'}}\left(\Omega \right)$ telles que:$$u\in L^2\left(\Omega \right);X_{j}u\in L^2\left(\Omega \right),j=1,...,r.$$