Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums

We work in the setting of The Flat Torus: Standing Notation, used with a natural number $n$ satisfying $1\le n$; the exponent $p$ and the wrapping map of that setting are not used. Let $\pi$ be the real number of The Number Pi §pi, let $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ be the real vector space of coefficient families and $\mu_{k}=1+4\pi^{2}\lVert k\rVert^{2}$ ($k\in\mathbb{Z}^{n}$) the Fourier weights, as in Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families; each $\mu_{k}$ satisfies $1\le\mu_{k}$ by Summability of the Negative Powers of the Fourier Weights of the Torus §product, used with its $s$ taken to be $n$, and $\mu_{k}^{s}$ denotes the natural power, with $\mu_{k}^{1}=\mu_{k}$. The sets $\Gamma_{N}$ ($N\in\mathbb{N}$) are the cubes of The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points, and cube sums, cube-summability and lattice sums are those of Cube Sums of Families on the Integer Lattice. For $m\in\mathbb{N}$, $H^{-m}(\mathbb{T}^{n})$ is the Sobolev space of order $-m$ with norm $|\cdot|_{H^{-m}}$. For $k\in\mathbb{Z}^{n}$ let $e_{k}\in\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ be the family with $e_{k}(k)=1$ and $e_{k}(k')=0$ for $k'\ne k$. Then the following hold.

1. (Summable powers)¶ Let $s\in\mathbb{N}$ satisfy $n<2s$. Then the family $k\mapsto\tfrac{1}{\mu_{k}^{s}}$ is cube-summable.

2. (The inverse weights are not summable)¶ Suppose $2\le n$. Then the set of cube sums $\bigl\{\sum_{k\in\Gamma_{N}}\tfrac{1}{\mu_{k}}:N\in\mathbb{N}\bigr\}$ is not bounded above.

3. (Sobolev norms as lattice sums)¶ Let $m\in\mathbb{N}$ and $c\in\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$. Then $c\in H^{-m}(\mathbb{T}^{n})$ if and only if the family $k\mapsto\tfrac{c(k)^{2}}{\mu_{k}^{m}}$ is cube-summable, and in that case

$$|c|_{H^{-m}}^{2}=\sum_{k\in\mathbb{Z}^{n}}\frac{c(k)^{2}}{\mu_{k}^{m}} .$$

4. (Unit families)¶ For every $m\in\mathbb{N}$ and $k\in\mathbb{Z}^{n}$, $e_{k}\in H^{-m}(\mathbb{T}^{n})$ and $|e_{k}|_{H^{-m}}^{2}=\tfrac{1}{\mu_{k}^{m}}$.