Periodic Convolution and Mollification on the Torus

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$ and a real number $p$ with $1\le p$; the cell $Q$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$ and the class $\mathcal{L}^{p}(\mathbb{T}^{n})$ are the ones fixed there, and $\mathcal{B}(\mathbb{R}^{n})$, $\lambda_{n}$ and the Euclidean norm $\lVert\,\cdot\,\rVert$ are those of the Borel $\sigma$-algebra, Lebesgue measure and Euclidean space. Let the periodic extension $\tilde{v}$ of a map $v:Q\to\mathbb{R}$ be as defined there. A map $\mathbb{R}^{n}\to\mathbb{R}$ is called continuous when it is continuous from $\mathbb{R}^{n}$ with its Euclidean distance to $\mathbb{R}$ with the metric of The Absolute Value Metric on the Real Numbers.

Let $R\in\mathbb{R}$ with $0<R$, let $\psi:\mathbb{R}^{n}\to\mathbb{R}$ be continuous and such that $\psi(y)=0$ for every $y\in\mathbb{R}^{n}$ with $R<\lVert y\rVert$, and let $u\in\mathcal{L}^{p}(\mathbb{T}^{n})$. Then $u$ is measurable with respect to $\mathcal{B}_{Q}$, being a $p$-integrable function on $(Q,\mathcal{B}_{Q},\lambda_{Q})$, and is integrable with respect to $\lambda_{Q}$ by The Periodic Extension of a Function on the Unit Cell (iv); hence $\tilde{u}$ is measurable with respect to $\mathcal{B}(\mathbb{R}^{n})$ by The Periodic Extension of a Function on the Unit Cell (i) and $\mathbf{1}_{B}\tilde{u}$ is integrable with respect to $\lambda_{n}$ for every bounded $B\in\mathcal{B}(\mathbb{R}^{n})$ by The Periodic Extension of a Function on the Unit Cell (v). So the pair $(\psi,\tilde{u})$ satisfies the hypotheses placed on the kernel and the function in Convolution of a Locally Integrable Function with a Compactly Supported Kernel.

1. (Periodic convolution)(i) The periodic convolution of $\psi$ with $u$ is the map $\psi\star u:\mathbb{R}^{n}\to\mathbb{R}$ given by the convolution of $\psi$ with the periodic extension $\tilde{u}$, namely

$$(\psi\star u)(x)=\int_{\mathbb{R}^{n}}\psi(x-y)\,\tilde{u}(y)\,d\lambda_{n}(y)\qquad(x\in\mathbb{R}^{n}),$$

a real number at every $x$ by the claim just cited.

2. (Mollification)(ii) Let $\delta,\varepsilon\in\mathbb{R}$ with $0<\delta$ and $0<\varepsilon$, let $\rho$ be a mollifier kernel of radius $\delta$ on $\mathbb{R}^{n}$, and let $\rho_{\varepsilon}$ be the rescaling of $\rho$ by $\varepsilon$, which is a mollifier kernel of radius $\varepsilon\delta$ on $\mathbb{R}^{n}$ by that lemma. By the smoothness and support conditions of Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$, together with claim 3 of Euclidean Space Is Open in Itself, the map $\rho_{\varepsilon}$ is continuous and vanishes at every $y$ with $\varepsilon\delta<\lVert y\rVert$, so it may be taken as the kernel $\psi$ above with $R=\varepsilon\delta$. The mollification of $u$ of parameter $\varepsilon$ by $\rho$ is the periodic convolution $\rho_{\varepsilon}\star u$.