Properties of Periodic Convolution 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 $(Q,\mathcal{B}_{Q},\lambda_{Q})$, the classes $\mathcal{L}^{p}(\mathbb{T}^{n})$ and $L^{p}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, the periodic classes $C_{\mathrm{per}}$, $C^{k}_{\mathrm{per}}$ and $C^{\infty}_{\mathrm{per}}$, and the restriction $v|_{Q}$ are the ones fixed there, as are $\mathcal{B}(\mathbb{R}^{n})$, $\lambda_{n}$ and the Euclidean norm $\lVert\,\cdot\,\rVert$. Let $\lVert\,\cdot\,\rVert_{r}$ denote the $L^{r}$ seminorm of $(Q,\mathcal{B}_{Q},\lambda_{Q})$, for a real number $r$ with $1\le r$; let the periodic extension $\tilde{v}$ be as defined there; and let $\psi\star u$ denote the periodic convolution. Continuity and smoothness of a map on $\mathbb{R}^{n}$ are read as in Periodic Convolution and Mollification on the Torus, and the partial derivatives $\partial_{i}$ and the classes $C^{k}$ on a Euclidean open set are those of the ambient calculus setting.

Throughout, $R$ is a real number with $0<R$, the map $\psi:\mathbb{R}^{n}\to\mathbb{R}$ is continuous with $\psi(y)=0$ for every $y\in\mathbb{R}^{n}$ satisfying $R<\lVert y\rVert$, and $u\in\mathcal{L}^{p}(\mathbb{T}^{n})$. Then the following hold.

1. (Periodicity and smoothness)(i) $\psi\star u\in C_{\mathrm{per}}$. Let $k$ be a natural number and suppose $\psi$ is of class $C^{k}$ on $\mathbb{R}^{n}$; then $\psi$ is of class $C^{1}$ on $\mathbb{R}^{n}$ by claim 2 of Euclidean Space Is Open in Itself. Then $\partial_{i}\psi$ is continuous and vanishes at every $y$ with $R<\lVert y\rVert$ for each $i\in[n]$, by Convolution of a Locally Integrable Function with a Compactly Supported Kernel (iv), so that $(\partial_{i}\psi)\star u$ is again a periodic convolution in the sense of Periodic Convolution and Mollification on the Torus (i); moreover $\psi\star u\in C^{k}_{\mathrm{per}}$ and

$$\partial_{i}(\psi\star u)=(\partial_{i}\psi)\star u\qquad\text{for every }i\in[n].$$

If $\psi$ is smooth, then $\psi\star u\in C^{\infty}_{\mathrm{per}}$.

2. (Dependence on the class only)(ii) Let $u'\in\mathcal{L}^{p}(\mathbb{T}^{n})$ satisfy $[u']=[u]$ in $L^{p}(\mathbb{T}^{n})$. Then $\psi\star u'=\psi\star u$.

3. (Linearity)(iii) Let $u'\in\mathcal{L}^{p}(\mathbb{T}^{n})$, let $\psi':\mathbb{R}^{n}\to\mathbb{R}$ be continuous with $\psi'(y)=0$ for every $y$ with $R<\lVert y\rVert$, and let $c\in\mathbb{R}$. Then $u+c\,u'\in\mathcal{L}^{p}(\mathbb{T}^{n})$ by Minkowski's Inequality (ii), the map $\psi+c\,\psi'$ is continuous and vanishes at every $y$ with $R<\lVert y\rVert$, and

$$\psi\star(u+c\,u')=\psi\star u+c\,(\psi\star u'),\qquad (\psi+c\,\psi')\star u=\psi\star u+c\,(\psi'\star u).$$

4. (A uniform bound)(iv) There is a real number $C$ with $0\le C$, depending only on $n$, $p$, $R$ and $\psi$, such that for every $u\in\mathcal{L}^{p}(\mathbb{T}^{n})$,

$$\bigl|(\psi\star u)(x)\bigr|\le C\,\lVert u\rVert_{p}\qquad\text{for every }x\in\mathbb{R}^{n};$$

consequently $(\psi\star u)|_{Q}\in\mathcal{L}^{p}(\mathbb{T}^{n})$ and $\bigl\lVert(\psi\star u)|_{Q}\bigr\rVert_{p}\le C\,\lVert u\rVert_{p}$.

5. (Uniform approximation of a continuous periodic function)(v) Let $\delta\in\mathbb{R}$ with $0<\delta$, let $\rho$ be a mollifier kernel of radius $\delta$ on $\mathbb{R}^{n}$, and let $w\in C_{\mathrm{per}}$, so that $w|_{Q}\in\mathcal{L}^{p}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i). Let $\eta\in\mathbb{R}$ with $0\le\eta$ be such that $|w(x)-w(z)|\le\eta$ for all $x,z\in\mathbb{R}^{n}$ with $\lVert x-z\rVert\le\delta$. Then

$$\bigl|\bigl(\rho\star(w|_{Q})\bigr)(x)-w(x)\bigr|\le\eta\qquad\text{for every }x\in\mathbb{R}^{n}.$$