Elementary Properties of Lattice-Periodic Functions

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background and in the setting of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, both used here with a natural number $n$ satisfying $1\le n$, exactly as in Lattice-Periodic Functions and the Periodic Function Classes, whose reading of continuity, of the classes $C^{k}$ on $\mathbb{R}^{n}$ and of smoothness on $\mathbb{R}^{n}$ is in force. Let $\mathbb{Z}^{n}$ be the integer lattice, let $\mathbb{Z}^{n}$-periodicity be as defined there, and let $C_{\mathrm{per}}$, $C^{k}_{\mathrm{per}}$ and $C^{\infty}_{\mathrm{per}}$ abbreviate the periodic function classes $C_{\mathrm{per}}(\mathbb{R}^{n})$, $C^{k}_{\mathrm{per}}(\mathbb{R}^{n})$ and $C^{\infty}_{\mathrm{per}}(\mathbb{R}^{n})$. Let $Q$, $\mathring{Q}$ and $\overline{Q}$ be the half-open, open and closed unit cells and let $\pi$ be the wrapping map of that lemma. For maps $u,v:\mathbb{R}^{n}\to\mathbb{R}$ and $c\in\mathbb{R}$, the maps $u+v$, $cu$ and $uv$ are the pointwise sum, scalar multiple and product of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions. Then the following hold.

1. (Algebraic closure)(i) Let $u,v:\mathbb{R}^{n}\to\mathbb{R}$ be $\mathbb{Z}^{n}$-periodic and let $c\in\mathbb{R}$. Then $u+v$, $cu$ and $uv$ are $\mathbb{Z}^{n}$-periodic. If in addition $u$ and $v$ both lie in $C_{\mathrm{per}}$, or both lie in $C^{k}_{\mathrm{per}}$ for a natural number $k$, or both lie in $C^{\infty}_{\mathrm{per}}$, then $u+v$, $cu$ and $uv$ lie in that same class.

2. (Derivatives of a periodic function)(ii) Let $u:\mathbb{R}^{n}\to\mathbb{R}$ be $\mathbb{Z}^{n}$-periodic, let $i\in[n]$, and suppose the partial derivative $\partial_{i}u(x)$ exists at every $x\in\mathbb{R}^{n}$. Then $\partial_{i}u:\mathbb{R}^{n}\to\mathbb{R}$ is $\mathbb{Z}^{n}$-periodic. Consequently, for every $i\in[n]$: if $u\in C^{1}_{\mathrm{per}}$ then $\partial_{i}u\in C_{\mathrm{per}}$; if $u\in C^{k+1}_{\mathrm{per}}$ for a natural number $k$ then $\partial_{i}u\in C^{k}_{\mathrm{per}}$; and if $u\in C^{\infty}_{\mathrm{per}}$ then $\partial_{i}u\in C^{\infty}_{\mathrm{per}}$.

3. (Boundedness)(iii) Let $u\in C_{\mathrm{per}}$. Then there is $y\in\overline{Q}$ such that $|u(x)|\le|u(y)|$ for every $x\in\mathbb{R}^{n}$. In particular there is a nonnegative real number $M$ with $|u(x)|\le M$ for every $x\in\mathbb{R}^{n}$.

4. (Periodisation)(iv) Let $W\subseteq\mathring{Q}$ and let $g:\mathbb{R}^{n}\to\mathbb{R}$ be continuous and such that $g(x)=0$ for every $x\in\mathbb{R}^{n}$ with $x\notin W$. Then the composite $g\circ\pi$, whose value at $x$ is $g(\pi(x))$, belongs to $C_{\mathrm{per}}$ and satisfies $g(\pi(x))=g(x)$ for every $x\in Q$. If moreover $0\le g(y)\le1$ for every $y\in\mathbb{R}^{n}$, then $0\le g(\pi(x))\le1$ for every $x\in\mathbb{R}^{n}$.