I am reading the paper "On Convolutions" (1958) and have encountered the notion of "Admissible" and "Permitted" spaces.


In p.17-18 of the above paper, it says that an admissible space $E$ is a locally convex space of distributions with $\mathcal{D} \subset E \subset \mathcal{D}'$ with


1. the inclusions being continuous
2. $\mathcal{D}$ is dense in $E$.


Also, for an admissible space $E$, it further defines the notion of "Permitted" by introducing regularizations $\{ \rho\_k \}$ and multiplicators $\{ \alpha\_k \}$. Here,


1. $\{ \rho\_k \}$ is a sequence of nonnegative, even functions with $\int \rho\_k=1$ and support tending to $\{0\}$ in $\mathbb{R}^n$.
2. $0 \leq \alpha\_k \leq 1$, $\alpha\_k \to 1$ in $\mathcal{E}$ and $\{ \alpha\_k \}$ is bounded in $\mathfrak{B}$.
3. Then, $E$ is permitted if $\alpha\_k (T \* \rho\_k) \to T$ and $(\alpha\_k T) \* \rho\_k) \to T$ in $E$ for all $T \in E$.


In the above 2, I was not able to understand what "bounded in $\mathfrak{B}$" exactly means. So, I looked for the cited reference but it is in French...


<http://sites.mathdoc.fr/OCLS/pdf/OCLS_1950__14__220_0.pdf>


Could anyone please explain to me what exactly the multiplicator is? Also, according to the above definitions, does $C^\infty( \mathbb{R}^n, \mathbb{R})$ with the Frechet topology qualify as the permitted space?



The notion of "Admissible" and "Permitted" in the context of convolution with distributions and hypocontinuity
The authors Yoshinaga and Ogata of *On concolutions* use Schwartz's notation for spaces of functions and distributions. The requirements for the sequence of smooth functions $\alpha\_k$ with values in $[0,1]$ are convergence to the constant function $1$ in the space $(\mathcal E)$ of all smooth functions, which means $\alpha\_k\to 1$ and $\partial^\beta\alpha\_k\to 0$ uniformly on compact sets for all multi-indices $\beta\neq 0$, as well as boundedness in the space $(\mathcal B)$ of all smooth functions with all derivatives bounded in $\mathbb R^n$ (this space $(\mathcal B)$ is introduced on page 199 of Schwartz's *Théorie des Distributions*), which means $$\sup\{|\partial^\beta \alpha\_k(x)|: x\in\mathbb R^n, k\in\mathbb N\}<\infty$$ for each multi-index $\beta$.


