The White-Noise Embedding of Lattice Fields: Linearity, Inversion by the Site Field, the Site Sum as the White-Noise Norm, and Coordinates

In the setting of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation, with $X=H^{-m}(\mathbb{T}^{n})$, its inner product $\langle\cdot,\cdot\rangle_{H^{-m}}$, its orthonormal basis $(e_{j})_{j\in\mathbb{N}}$, the noise weights $a_{j}$, the enumeration $\kappa$ of $\mathbb{Z}^{n}$ and the values $x(k)$ of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §data, let $M\in\mathbb{N}$. The noise space $X^{a}$ and its noise norm $|\cdot|_{a}$ are those of White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §noise-space. Let $L$, $\Gamma_{M}$, $\psi_{k}$, $\mathbb{L}_{M}$, lattice fields, their real vector space $\mathrm{Map}(\mathbb{L}_{M},\mathbb{R})$ and their discrete Fourier coefficients $\hat{f}(k)$ be those of The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field, and let the white-noise embedding $\mathcal{E}_{M}$, the site field $\varphi_{x}$ of $x\in X$ and $L^{\pm n/2}$ be those of The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution. Let $d_{M}\in\mathbb{N}$ and $\chi_{M}:[d_{M}]\to\{0,1\}$ be the Galerkin head dimension and the indicator of the active indices of The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §head-dimension, read with $M$ in place of $N$. For $j\in\mathbb{N}$, $a_{j}^{1/2}$ and $a_{j}^{-1/2}$ are those of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §heads, and for $x\in X$, $x_{j}=\langle x,e_{j}\rangle_{H^{-m}}$ is the coordinate of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates. Sums over finite index sets are those of Sum over a Finite Index Set.

1. (Linear and injective)¶ $\mathcal{E}_{M}$ is linear and injective: for lattice fields $f$ and $g$, $\mathcal{E}_{M}f=\mathcal{E}_{M}g$ only if $f=g$.

2. (Left inverse)¶ $\varphi_{\mathcal{E}_{M}f}=f$ for every lattice field $f$.

3. (Site evaluations)¶ For every $z\in\mathbb{L}_{M}$ the map $x\mapsto\varphi_{x}(z)$ from $X$ to $\mathbb{R}$ is linear and continuous.

4. (Restriction to the cube)¶ For every $x\in X$, $\mathcal{E}_{M}\varphi_{x}$ is the coefficient family equal to $x(k)$ at $k\in\Gamma_{M}$ and to $0$ at $k\in\mathbb{Z}^{n}$ with $k\notin\Gamma_{M}$.

5. (The site sum is the white-noise norm)¶ For every lattice field $f$, $\mathcal{E}_{M}f\in X^{a}$ and

$$|\mathcal{E}_{M}f|_{a}^{2}=\sum_{z\in\mathbb{L}_{M}}f(z)^{2}.$$

6. (The site bound)¶ For every $h\in X^{a}$,

$$\sum_{z\in\mathbb{L}_{M}}\varphi_{h}(z)^{2}=\sum_{k\in\Gamma_{M}}h(k)^{2}\le|h|_{a}^{2}.$$

7. (Coordinates)¶ For every $x\in X$ and $j\in\mathbb{N}$, $x_{j}=a_{j}^{1/2}\,x(\kappa(j))$, and for every $z\in\mathbb{L}_{M}$

$$\varphi_{x}(z)=L^{-n/2}\sum_{j=1}^{d_{M}}\chi_{M}(j)\,a_{j}^{-1/2}\,x_{j}\,\psi_{\kappa(j)}(z).$$