Existence of Mollifier Kernels of Every Radius

Let $n$ be a natural number with $1\le n$, let $\mathbb{R}$ be the real numbers, let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on Euclidean space $\mathbb{R}^{n}$, and let $d_{E}$ be the Euclidean distance, a metric on $\mathbb{R}^{n}$; by The Open Subsets of a Metric Space Form a Topology the subsets open in $(\mathbb{R}^{n},d_{E})$ form a topology on $\mathbb{R}^{n}$, to which the topological notions below refer. Let $\lambda_{n}$ be Lebesgue measure on the Borel $\sigma$-algebra of $\mathbb{R}^{n}$. For $t\in\mathbb{R}$ write $t^{2}$ for $t\cdot t$, and write $t^{-1}$ for the multiplicative inverse of $t\ne 0$.

Let $\delta\in\mathbb{R}$ with $0<\delta$, let $\varphi:\mathbb{R}\to\mathbb{R}$ be the function of The Exponential Bump Building Block is Smooth on the Real Line, and let $\rho_{0}:\mathbb{R}^{n}\to\mathbb{R}$ be given by

$$\rho_{0}(y)=\varphi\bigl(\delta^{2}-\lVert y\rVert^{2}\bigr)\qquad(y\in\mathbb{R}^{n}).$$

Then the following hold.

1. (Properties of $\rho_{0}$) $\rho_{0}$ is smooth on $\mathbb{R}^{n}$; $0\le\rho_{0}(y)$ for every $y\in\mathbb{R}^{n}$; $\rho_{0}(y)=0$ for every $y$ with $\delta<\lVert y\rVert$; $\rho_{0}$ is compactly supported and integrable with respect to $\lambda_{n}$; and

$$0<\int_{\mathbb{R}^{n}}\rho_{0}\,d\lambda_{n}<\infty .$$

2. (Normalisation) Let $c$ be the multiplicative inverse of $\int_{\mathbb{R}^{n}}\rho_{0}\,d\lambda_{n}$, and let $\rho:\mathbb{R}^{n}\to\mathbb{R}$ be given by $\rho(y)=c\,\rho_{0}(y)$. Then $\rho$ is a mollifier kernel of radius $\delta$ on $\mathbb{R}^{n}$. In particular, for every natural number $n$ with $1\le n$ and every real $\delta>0$ a mollifier kernel of radius $\delta$ on $\mathbb{R}^{n}$ exists.