Proof of Elementary Properties of Lattice-Periodic Functions

Each result cited below is universally quantified over the data appearing in its own statement, and is applied to the data named here. Throughout, $\mathbb{R}^{n}$ is open in $\mathbb{R}^{n}$ by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, so results about $C^{k}$ maps on a Euclidean open set apply to it.

Claim 1. For $x\in\mathbb{R}^{n}$ and $m\in\mathbb{Z}^{n}$,

and in the same way $(cu)(x+m)=c\,u(x+m)=c\,u(x)=(cu)(x)$ and $(uv)(x+m)=u(x+m)v(x+m)=u(x)v(x)=(uv)(x)$. So all three maps are $\mathbb{Z}^{n}$-periodic. If $u$ and $v$ are continuous on $\mathbb{R}^{n}$, then so are $u+v$, $cu$ and $uv$ by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, applied with the metric space $(\mathbb{R}^{n},d_{E})$ and $A=\mathbb{R}^{n}$. If $u$ and $v$ are of class $C^{k}$ on $\mathbb{R}^{n}$, or both smooth on $\mathbb{R}^{n}$, then so are $u+v$, $cu$ and $uv$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. Combining with periodicity gives the assertion for each of the three classes.

Claim 2. Fix $m\in\mathbb{Z}^{n}$, $x\in\mathbb{R}^{n}$ and $i\in[n]$. For a real $t$ let $x\oplus t$ denote the point obtained from $x$ by replacing its $i$th coordinate by $x_{i}+t$; then $(x+m)\oplus t=(x\oplus t)+m$, since adding $m$ changes each coordinate by the corresponding coordinate of $m$ and the two operations act independently. By Partial Derivative on a Euclidean Open Set the number $\partial_{i}u(x+m)$ is the limit as $t$ tends to $0$ of

the last equality by $\mathbb{Z}^{n}$-periodicity of $u$. The difference quotients defining $\partial_{i}u(x+m)$ and $\partial_{i}u(x)$ therefore coincide for every $t\ne0$, so the two derivatives exist together and are equal. Hence $\partial_{i}u$ is $\mathbb{Z}^{n}$-periodic.

Now let $u\in C^{1}_{\mathrm{per}}$. By clause 1 of C^k Maps on a Euclidean Open Set, read through the scalar convention of clause 3 there, the partial derivative $\partial_{i}u$ exists at every point and $\partial_{i}u$ is continuous on $\mathbb{R}^{n}$; with the periodicity just proved, $\partial_{i}u\in C_{\mathrm{per}}$. If $u\in C^{k+1}_{\mathrm{per}}$ for a natural number $k$, then $\partial_{i}u$ is of class $C^{k}$ on $\mathbb{R}^{n}$ by clause 2 of C^k Maps on a Euclidean Open Set, so $\partial_{i}u\in C^{k}_{\mathrm{per}}$. If $u\in C^{\infty}_{\mathrm{per}}$, then $u$ is of class $C^{k+1}$ for every natural number $k$ by Smooth Map on a Euclidean Open Set, so $\partial_{i}u$ is of class $C^{k}$ for every natural number $k$, that is, smooth; hence $\partial_{i}u\in C^{\infty}_{\mathrm{per}}$.

Claim 3. The set $\overline{Q}$ is compact by The Half-Open Unit Cell Tiles Euclidean Space §cell and nonempty, since the origin lies in it. The restriction of $u$ to $\overline{Q}$ is continuous relative to $\overline{Q}$: given $z\in\overline{Q}$ and a positive $\varepsilon$, the $\delta$ furnished by continuity of $u$ at $z$ relative to $\mathbb{R}^{n}$ works verbatim for the smaller set, by Continuous Map Between Metric Spaces. By Extreme Value Theorem on a Compact Subset of a Metric Space there are $z_{-},z_{+}\in\overline{Q}$ with $u(z_{-})\le u(z)\le u(z_{+})$ for every $z\in\overline{Q}$. Let $y$ be whichever of $z_{-},z_{+}$ has $|u(z_{-})|\le|u(y)|$ and $|u(z_{+})|\le|u(y)|$, that is, the one with the larger absolute value of $u$. For $z\in\overline{Q}$ we then get

the outer bounds by claim 3 of Properties of the Absolute Value in an Ordered Field together with $|u(z_{\pm})|\le|u(y)|$, so $|u(z)|\le|u(y)|$ by claim 6 there.

Now let $x\in\mathbb{R}^{n}$ be arbitrary. By The Half-Open Unit Cell Tiles Euclidean Space §tiling there is $m\in\mathbb{Z}^{n}$ with $x-m\in Q$, and then $x=(x-m)+m$, so $u(x)=u(x-m)$ by periodicity. Since $x-m\in Q\subseteq\overline{Q}$ by The Half-Open Unit Cell Tiles Euclidean Space §cell, the previous paragraph gives $|u(x)|=|u(x-m)|\le|u(y)|$. Taking $M=|u(y)|$, which is nonnegative by claim 1 of Properties of the Absolute Value in an Ordered Field, completes the proof.

Claim 4. Periodicity and the values on $Q$. For $x\in\mathbb{R}^{n}$ and $k\in\mathbb{Z}^{n}$ we have $\pi(x+k)=\pi(x)$ by The Half-Open Unit Cell Tiles Euclidean Space §wrap, so $g(\pi(x+k))=g(\pi(x))$ and $g\circ\pi$ is $\mathbb{Z}^{n}$-periodic. For $x\in Q$ the same clause gives $\pi(x)=x$, so $g(\pi(x))=g(x)$. If $0\le g(y)\le1$ for every $y$, then in particular $0\le g(\pi(x))\le1$ for every $x$.

A local finite sum. Fix $x^{0}\in\mathbb{R}^{n}$ and put

where $\lfloor\,\cdot\,\rfloor$ is the integer part. For each $i$ the integers $m_{i}$ allowed lie between $\lfloor x^{0}_{i}\rfloor-1$ and $\lfloor x^{0}_{i}\rfloor+1$, and by claim 3 of Arithmetic, Order and Discreteness of the Integers such an integer equals one of $\lfloor x^{0}_{i}\rfloor-1$, $\lfloor x^{0}_{i}\rfloor$, $\lfloor x^{0}_{i}\rfloor+1$; so $J$ is contained in the set of $n$-tuples whose entries range over three-element sets, which is finite by Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets, and $J$ itself is finite by claim 3 of Basic Properties of Finite Sets. Fix an enumeration $m^{(1)},\dots,m^{(N)}$ of $J$ and define

$G$ is continuous. For fixed $m\in\mathbb{Z}^{n}$ the map $x\mapsto g(x-m)$ is continuous on $\mathbb{R}^{n}$: given $x^{1}$ and a positive $\varepsilon$, continuity of $g$ at $x^{1}-m$ provides a positive $\delta$ such that $d_{E}(z,x^{1}-m)<\delta$ implies $|g(z)-g(x^{1}-m)|<\varepsilon$, and $d_{E}(x-m,x^{1}-m)=\lVert(x-m)-(x^{1}-m)\rVert=\lVert x-x^{1}\rVert=d_{E}(x,x^{1})$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, so the same $\delta$ serves in Continuous Map Between Metric Spaces. A sum of $N$ continuous real-valued maps on $\mathbb{R}^{n}$ is continuous, by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space applied $N-1$ times.

$G$ agrees with $g\circ\pi$ near $x^{0}$. Let $x\in\mathbb{R}^{n}$ with $\lVert x-x^{0}\rVert<\tfrac12$. We first record which integer vectors can matter. Suppose $m\in\mathbb{Z}^{n}$ satisfies $x-m\in\mathring{Q}$, that is $0<x_{i}-m_{i}<1$ for every $i$. Then $x_{i}-1<m_{i}<x_{i}$, and $|x_{i}-x^{0}_{i}|\le\lVert x-x^{0}\rVert<\tfrac12$ by claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, so $x^{0}_{i}-\tfrac32<m_{i}<x^{0}_{i}+\tfrac12$. Since $\lfloor x^{0}_{i}\rfloor\le x^{0}_{i}<\lfloor x^{0}_{i}\rfloor+1$, this gives $\lfloor x^{0}_{i}\rfloor-\tfrac32<m_{i}<\lfloor x^{0}_{i}\rfloor+\tfrac32$, hence $\lfloor x^{0}_{i}\rfloor-1\le m_{i}\le\lfloor x^{0}_{i}\rfloor+1$ because $m_{i}$ is an integer and claim 3 of Arithmetic, Order and Discreteness of the Integers excludes integers strictly between consecutive ones. So $m\in J$.

Now suppose some summand of $G(x)$ is nonzero, say $g(x-m)\ne0$ with $m\in J$. Then $x-m\in W\subseteq\mathring{Q}\subseteq Q$, so by the uniqueness in The Half-Open Unit Cell Tiles Euclidean Space §tiling the vector $m$ is the one attached to $x$ and $x-m=\pi(x)$; the same uniqueness shows no other $m'\in\mathbb{Z}^{n}$ has $x-m'\in Q$, hence no other summand is nonzero, and $G(x)=g(\pi(x))$. Suppose instead every summand of $G(x)$ vanishes, so $G(x)=0$. If $g(\pi(x))\ne0$ then $\pi(x)\in W\subseteq\mathring{Q}$, and with $m=x-\pi(x)\in\mathbb{Z}^{n}$ we get $x-m\in\mathring{Q}$, hence $m\in J$ by the previous paragraph and $g(x-m)=g(\pi(x))\ne0$, contradicting the assumption. So $g(\pi(x))=0=G(x)$. In both cases $G(x)=g(\pi(x))$.

Continuity of $g\circ\pi$ at $x^{0}$. Let $\varepsilon$ be positive. Continuity of $G$ at $x^{0}$ gives a positive $\delta_{0}$ with $|G(x)-G(x^{0})|<\varepsilon$ whenever $d_{E}(x,x^{0})<\delta_{0}$. Let $\delta$ be the lesser of $\delta_{0}$ and $\tfrac12$, positive by claim 9 of Elementary Order Arithmetic in an Ordered Field. If $d_{E}(x,x^{0})<\delta$ then $\lVert x-x^{0}\rVert<\tfrac12$ and $\lVert x^{0}-x^{0}\rVert=0<\tfrac12$, so the previous paragraph gives $G(x)=g(\pi(x))$ and $G(x^{0})=g(\pi(x^{0}))$, whence $|g(\pi(x))-g(\pi(x^{0}))|<\varepsilon$. Thus $g\circ\pi$ is continuous at $x^{0}$ relative to $\mathbb{R}^{n}$, and since $x^{0}$ was arbitrary it is continuous on $\mathbb{R}^{n}$. Together with the periodicity established above, $g\circ\pi\in C_{\mathrm{per}}$.