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

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$, and in the setting of Real Hilbert Spaces: Standing Notation and Background, whose standing space $H$ is not used here. The real Hilbert space $L^{2}(\mathbb{T}^{n})$ with inner product $\langle\cdot,\cdot\rangle_{L^{2}}$ and norm $\lVert\cdot\rVert_{L^{2}}$, the coefficient families, the Fourier coefficient family $\hat{U}$ of a class $U$, the classes $E_{k}$, the enumerations of the lattice $\mathbb{Z}^{n}$, the Fourier weights $\mu_{k}=1+4\pi^{2}\lVert k\rVert^{2}$ and the positive numbers $\rho_{k}$ with $\rho_{k}^{2}=\frac{1}{\mu_{k}}$, and the natural powers $t^{l}$, are as in Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families. Let $m\in\mathbb{N}$ satisfy $n\le m+1$, and let $\kappa$ be an enumeration of $\mathbb{Z}^{n}$. $H^{-m}(\mathbb{T}^{n})$ is the Sobolev space of order $-m$, with the inner product $\langle\cdot,\cdot\rangle_{H^{-m}}$; it is a real Hilbert space by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale §hilbert; for $\Phi\in H^{-m}(\mathbb{T}^{n})$ and $k\in\mathbb{Z}^{n}$, $\Phi(k)$ is the value of the coefficient family $\Phi$ at $k$; and $\zeta_{m,k}\in H^{-m}(\mathbb{T}^{n})$ are the elements of The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale §basis. For $j\in\mathbb{N}$ put

$$e_{j}=\zeta_{m,\kappa(j)},\qquad a_{j}=\bigl(\rho_{\kappa(j)}^{m}\bigr)^{2},\qquad c_{j}=\bigl(\rho_{\kappa(j)}^{m+1}\bigr)^{2}.$$

1. (The basis)¶ $(e_{j})_{j\in\mathbb{N}}$ is an orthonormal basis of $H^{-m}(\mathbb{T}^{n})$, and $\langle\Phi,e_{j}\rangle_{H^{-m}}=\rho_{\kappa(j)}^{m}\,\Phi(\kappa(j))$ for every $\Phi\in H^{-m}(\mathbb{T}^{n})$ and $j\in\mathbb{N}$.

2. (The noise weights)¶ $a=(a_{j})_{j\in\mathbb{N}}$ is a weight sequence, and $a_{j}\le1$ for every $j\in\mathbb{N}$.

3. (The variances)¶ $c=(c_{j})_{j\in\mathbb{N}}$ is a variance sequence.

4. (The ratio)¶ For every $j\in\mathbb{N}$, $c_{j}\le a_{j}$ and $a_{j}=\mu_{\kappa(j)}\,c_{j}$.

5. (The noise space is the square-integrable space)¶ Read The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis with the real Hilbert space $H^{-m}(\mathbb{T}^{n})$ in place of $X$, the orthonormal basis $(e_{j})_{j\in\mathbb{N}}$ of claim 1 and the weight sequence $a$ of claim 2, and let $X^{a}$, $\langle\cdot,\cdot\rangle_{a}$ and $|\cdot|_{a}$ be the noise space, noise pairing and noise norm so obtained. Then for every $U\in L^{2}(\mathbb{T}^{n})$ one has $\hat{U}\in X^{a}$, and the map $U\mapsto\hat{U}$ is a bijection from $L^{2}(\mathbb{T}^{n})$ onto $X^{a}$ with

$$\langle\hat{U},\hat{U}'\rangle_{a}=\langle U,U'\rangle_{L^{2}}\qquad\text{for all }U,U'\in L^{2}(\mathbb{T}^{n}).$$

6. (The noise basis is the trigonometric basis)¶ For every $j\in\mathbb{N}$, $a_{j}^{1/2}=\rho_{\kappa(j)}^{m}$, where $a_{j}^{1/2}$ is the nonnegative square root of $a_{j}$, and $a_{j}^{1/2}\,e_{j}=\hat{E}_{\kappa(j)}$ for every $j\in\mathbb{N}$.