Lattice-Periodic Functions and the Periodic Function Classes

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 $1\le n$. The two settings fix Euclidean space $\mathbb{R}^{n}$, the sum $x+y$ and difference $x-y$ of its points, the Euclidean norm $\lVert\,\cdot\,\rVert$ and distance $d_{E}$, and the notions of open, closed, bounded and compact subset, each by reference to the same definitions, so their readings agree and are used interchangeably below. From the first we take the integers $\mathbb{Z}$, the natural numbers $\mathbb{N}$ and the initial segments $[n]$; from the second, the partial derivatives $\partial_{i}$ and the classes $C^{k}$ on a Euclidean open set. The set $\mathbb{R}^{n}$ is open in $\mathbb{R}^{n}$ by claim 1 of Euclidean Space Is Open in Itself, so the calculus notions above apply to maps defined on all of $\mathbb{R}^{n}$. Smoothness on $\mathbb{R}^{n}$ is that of Smooth Functions on an Open Subset of ℝⁿ, and a map $u:\mathbb{R}^{n}\to\mathbb{R}$ is called continuous when it is continuous on $\mathbb{R}^{n}$ as a map from $(\mathbb{R}^{n},d_{E})$ to $\mathbb{R}$ with the metric of The Absolute Value Metric on the Real Numbers.

1. (The integer lattice)(i) $\mathbb{Z}^{n}$ denotes the set of those points of $\mathbb{R}^{n}$ all of whose coordinates are integers,

$$\mathbb{Z}^{n}=\{m\in\mathbb{R}^{n}\ :\ m_{i}\in\mathbb{Z}\ \text{for every}\ i\in[n]\}.$$

2. (Periodicity)(ii) A map $u:\mathbb{R}^{n}\to\mathbb{R}$ is $\mathbb{Z}^{n}$-periodic if

$$u(x+m)=u(x)\qquad\text{for every}\ x\in\mathbb{R}^{n}\ \text{and every}\ m\in\mathbb{Z}^{n}.$$

The same condition, read for a map $u:\mathbb{R}^{n}\to[0,\infty]$ into the extended half-line of Measure, Measure Space, and Probability Measure, defines $\mathbb{Z}^{n}$-periodicity of such a map.

3. (The periodic classes)(iii) $C_{\mathrm{per}}(\mathbb{R}^{n})$ denotes the set of $\mathbb{Z}^{n}$-periodic maps $u:\mathbb{R}^{n}\to\mathbb{R}$ that are continuous. For a natural number $k$, $C^{k}_{\mathrm{per}}(\mathbb{R}^{n})$ denotes the set of $\mathbb{Z}^{n}$-periodic maps $u:\mathbb{R}^{n}\to\mathbb{R}$ that are of class $C^{k}$ on $\mathbb{R}^{n}$, and $C^{\infty}_{\mathrm{per}}(\mathbb{R}^{n})$ denotes the set of $\mathbb{Z}^{n}$-periodic maps $u:\mathbb{R}^{n}\to\mathbb{R}$ that are smooth on $\mathbb{R}^{n}$.