The Periodised Kernel of a Periodic Convolution

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the cell $Q$, the lattice $\mathbb{Z}^{n}$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$, the integral over $\mathbb{T}^{n}$, the classes $\mathcal{L}^{r}(\mathbb{T}^{n})$ for a real $r$ with $1\le r$, the periodic classes $C_{\mathrm{per}}$ and $C^{\infty}_{\mathrm{per}}$, the restriction $v|_{Q}$, and $\mathcal{B}(\mathbb{R}^{n})$, $\lambda_{n}$ and the Euclidean norm $\lVert\,\cdot\,\rVert$ are the ones fixed there. A map $\mathbb{R}^{n}\to\mathbb{R}$ is called continuous, and smooth, as in Periodic Convolution and Mollification on the Torus; $\psi\star u$ denotes the periodic convolution; and a sum indexed by a finite set is that of Sum over a Finite Index Set.

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

1. (The kernel is integrable)(i) The map $\psi$ is compactly supported, bounded, and integrable with respect to $\lambda_{n}$. We write

$$K=\int_{\mathbb{R}^{n}}|\psi|\,d\lambda_{n},$$

a real number with $0\le K$.

2. (The periodised kernel)(ii) For all $x,y\in\mathbb{R}^{n}$ the set $S(x,y)=\{m\in\mathbb{Z}^{n}:\psi(x-y-m)\ne0\}$ is finite, and for every nonempty finite $F\subseteq\mathbb{Z}^{n}$ with $S(x,y)\subseteq F$ — for instance $F=S(x,y)\cup\{0\}$ — the sum

$$\sum_{m\in F}\psi(x-y-m)$$

has one and the same value. Writing $\Psi(x,y)$ for that common value defines a map $\Psi:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$, the periodised kernel of $\psi$.

3. (Regularity in each variable)(iii) Let $x\in\mathbb{R}^{n}$. Then the map $y\mapsto\Psi(x,y)$ belongs to $C_{\mathrm{per}}$, and belongs to $C^{\infty}_{\mathrm{per}}$ if $\psi$ is smooth. Symmetrically, for $y\in\mathbb{R}^{n}$ the map $x\mapsto\Psi(x,y)$ belongs to $C_{\mathrm{per}}$, and to $C^{\infty}_{\mathrm{per}}$ if $\psi$ is smooth.

4. (Mass bound over the cell)(iv) Let $x\in\mathbb{R}^{n}$ and let $\Psi_{x}$ be the restriction of $y\mapsto\Psi(x,y)$ to $Q$. Then $\Psi_{x}\in\mathcal{L}^{1}(\mathbb{T}^{n})$ and

$$\int_{\mathbb{T}^{n}}|\Psi_{x}|\,dy\le K .$$

Symmetrically, for $y\in\mathbb{R}^{n}$ the restriction $\Psi^{y}$ of $x\mapsto\Psi(x,y)$ to $Q$ lies in $\mathcal{L}^{1}(\mathbb{T}^{n})$ and satisfies the same bound.

5. (Representation of the periodic convolution)(v) Let $u\in\mathcal{L}^{1}(\mathbb{T}^{n})$ and let $x\in\mathbb{R}^{n}$. Then the pointwise product $\Psi_{x}\,u$ is integrable with respect to $\lambda_{Q}$ and

$$(\psi\star u)(x)=\int_{\mathbb{T}^{n}}\Psi_{x}\,u\,dy .$$

6. (Joint measurability over the cell)(vi) The restriction of $\Psi$ to $Q\times Q$ is measurable with respect to the product $\sigma$-algebra $\mathcal{B}_{Q}\otimes\mathcal{B}_{Q}$.

7. (Differentiating the periodised kernel)(vii) Suppose $\psi$ is smooth, and let $i\in[n]$. Then $\psi$ is of class $C^{1}$ on $\mathbb{R}^{n}$ by claim 2 of Euclidean Space Is Open in Itself, and the partial derivative $\partial_{i}\psi$ is continuous and vanishes at every $y\in\mathbb{R}^{n}$ with $R<\lVert y\rVert$, by Convolution of a Locally Integrable Function with a Compactly Supported Kernel (iv); so $\partial_{i}\psi$ satisfies the standing hypotheses placed on $\psi$ above, and claim 2 applied to it yields its periodised kernel, which we write $\Psi^{(i)}$. Let $x\in\mathbb{R}^{n}$ and write $\Psi(x,\cdot)$ for the map $y\mapsto\Psi(x,y)$, a member of $C^{\infty}_{\mathrm{per}}$ by claim 3. Then

$$\partial_{i}\Psi(x,\cdot)(y)=-\Psi^{(i)}(x,y)\qquad\text{for every }y\in\mathbb{R}^{n}.$$