Let $\tilde{f}$, $\tilde{g}$ be ultradistributions in ${𝒵}^{\text{'}}$ and let ${\tilde{f}}_n=\tilde{f}*{\delta }_n$ and ${\tilde{g}}_n=\tilde{g}*{\sigma }_n$ where $\left\{{\delta }_n\right\}$ is a sequence in $𝒵$ which converges to the Dirac-delta function $\delta$. Then the neutrix product $\tilde{f}\diamond \tilde{g}$ is defined on the space of ultradistributions ${𝒵}^{\text{'}}$ as the neutrix limit of the sequence $\left\{\frac{1}{2}({\tilde{f}}_n\tilde{g}+\tilde{f}{\tilde{g}}_n)\right\}$ provided the limit $\tilde{h}$ exist in the sense that $$\underset{n\to \infty }{\mathrm{N}\text{-}lim}\frac{1}{2}\langle {\tilde{f}}_n\tilde{g}+\tilde{f}{\tilde{g}}_n,\psi \rangle =\langle \tilde{h},\psi \rangle$$ for all $\psi$ in $𝒵$. We also prove that the neutrix convolution product $f♦*g$ exist in ${𝒟}^{\text{'}}$, if and only if the neutrix product $\tilde{f}\diamond \tilde{g}$ exist in ${𝒵}^{\text{'}}$ and the exchange formula $$F\left(f♦*g\right)=\tilde{f}\diamond \tilde{g}$$ is then satisfied.

In this paper, that is divided in two parts, we study the distributional Dunkl transform on R. In the first part we investigate the Dunkl transform and the Dunkl convolution operators on tempered distributions. We prove that the tempered distributions defining Dunkl convolution operators on the Schwartz space ƒ are the elements of ${𝒪}_c^{\prime }$, the space of usual convolution operators on $S$. In the second part we define the distributional Dunkl transform by employing the kernel method. We introduce...

Let ${𝒯}_X$ denote the operator-norm closure of the class of convolution operators ${\Phi }_{\mu }:X\to X$ where $X$ is a suitable function space on $R$. Let ${ℳ}_r^p$ be the closed subspace of regular functions in the Marinkiewicz space ${ℳ}^p$, $1\le p\<\infty$. We show that the space ${𝒯}_{{ℳ}_r^p}$ is isometrically isomorphic to ${𝒯}_{L^p}$ and that strong operator sequential convergence and norm convergence in ${𝒯}_{{ℳ}_r^p}$ coincide. We also obtain some results concerning convolution operators under the Wiener transformation. These are to improve a Tauberian theorem of Wiener...

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 $G$ be a locally compact group, for $p\in (1,\infty )$ let $P{f}_p\left(G\right)$ denote the closure of $L^1\left(G\right)$ in the convolution operators on $L^p\left(G\right)$. Denote $W_p\left(G\right)$ the dual of $P{f}_p\left(G\right)$ which is contained in the space of pointwise multipliers of the Figa-Talamanca Herz space $A_p\left(G\right)$. It is shown that on the unit sphere of $W_p\left(G\right)$ the $\sigma (W_p,P{f}_p)$ topology and the strong $A_p$-multiplier topology coincide.