Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families

Let $n\in\mathbb{N}$, let $\mathbb{Z}^{n}$ be the integer lattice with its cubes $\Gamma_{N}$, and let the cube sums $S_{N}$, cube-summability and the lattice sum $\sum_{k\in\mathbb{Z}^{n}}$ be as defined there. For a family $a:\mathbb{Z}^{n}\to\mathbb{R}$ write $|a|$ for the family $k\mapsto|a(k)|$. For a nonempty finite set $E\subseteq\mathbb{Z}^{n}$, $\sum_{k\in E}a(k)$ is the sum over the finite index set $E$; upper bounds and suprema of sets of real numbers are those of The Real Numbers: Standing Notation and Background §bounds. Let $a,b:\mathbb{Z}^{n}\to\mathbb{R}$ and $\lambda\in\mathbb{R}$. Then the following hold.

1. (Linearity)¶ If $a$ and $b$ are cube-summable, then so are $a+b$ and $\lambda a$ (pointwise operations), and

$$\sum_{k\in\mathbb{Z}^{n}}\bigl(a(k)+b(k)\bigr)=\sum_{k\in\mathbb{Z}^{n}}a(k)+\sum_{k\in\mathbb{Z}^{n}}b(k),\qquad\sum_{k\in\mathbb{Z}^{n}}\lambda a(k)=\lambda\sum_{k\in\mathbb{Z}^{n}}a(k).$$

2. (Nonnegative families)¶ Suppose $0\le a(k)$ for every $k\in\mathbb{Z}^{n}$. Then $S_{N}(a)\le S_{M}(a)$ whenever $N\le M$, and $a$ is cube-summable if and only if the set $\{S_{N}(a):N\in\mathbb{N}\}$ is bounded above; in that case $\sum_{k\in\mathbb{Z}^{n}}a(k)$ is the supremum of that set, $0\le\sum_{k\in\mathbb{Z}^{n}}a(k)$, and $\sum_{k\in E}a(k)\le\sum_{k\in\mathbb{Z}^{n}}a(k)$ for every nonempty finite set $E\subseteq\mathbb{Z}^{n}$.

3. (Enumerations)¶ Suppose $0\le a(k)$ for every $k\in\mathbb{Z}^{n}$, and let $\kappa:\mathbb{N}\to\mathbb{Z}^{n}$ be a bijection. Then the series $\sum_{j=1}^{\infty}a(\kappa(j))$ converges if and only if $a$ is cube-summable, and in that case

$$\sum_{j=1}^{\infty}a(\kappa(j))=\sum_{k\in\mathbb{Z}^{n}}a(k).$$

4. (Absolute summability)¶ If $|a|$ is cube-summable, then $a$ is cube-summable and $\bigl|\sum_{k\in\mathbb{Z}^{n}}a(k)\bigr|\le\sum_{k\in\mathbb{Z}^{n}}|a(k)|$.

5. (Comparison)¶ If $b$ is cube-summable and $|a(k)|\le b(k)$ for every $k\in\mathbb{Z}^{n}$, then $|a|$ and $a$ are cube-summable and $\sum_{k\in\mathbb{Z}^{n}}|a(k)|\le\sum_{k\in\mathbb{Z}^{n}}b(k)$.

6. (Finitely supported families)¶ If $E\subseteq\mathbb{Z}^{n}$ is a nonempty finite set and $a(k)=0$ for every $k\notin E$, then $a$ is cube-summable, $S_{N}(a)=\sum_{k\in E}a(k)$ for every $N$ with $E\subseteq\Gamma_{N}$, and $\sum_{k\in\mathbb{Z}^{n}}a(k)=\sum_{k\in E}a(k)$.