Proof 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

Each result cited is universally quantified over the data in its own statement. Elementary arithmetic and order facts about real numbers (commutativity and associativity of products, multiplication of an inequality by a nonnegative number, positivity of products and of inverses of positive numbers, $t\,t^{-1}=1$, $t^{2}=tt$ for natural powers, transitivity of $\le$, $1\cdot t=t$) are in force by the numbers clause of the real Hilbert space setting, which puts the background of the real-number setting in force. Throughout, $j\in\mathbb{N}$ and we write $k=\kappa(j)$. Since $\kappa$ is an enumeration, it is a bijection of $\mathbb{N}$ onto $\mathbb{Z}^{n}$, so in particular $\kappa(j)=\kappa(j')$ implies $j=j'$.

Preliminary: the Fourier weights. By Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families, $\mu_{k}$ is the weight of Summability of the Negative Powers of the Fourier Weights of the Torus, which applies with $s=m+1$ since $n\le m+1$; $1\le\mu_{k}$, and hence $\mu_{k}$ is positive, for every $k\in\mathbb{Z}^{n}$ by Summability of the Negative Powers of the Fourier Weights of the Torus §product.

Claim 1. By the Hilbert space clause of the negative Sobolev lemma, $H^{-m}(\mathbb{T}^{n})$ is a real Hilbert space. By its basis clause, applied with the present $m$ and the enumeration $\kappa$, the sequence $(\zeta_{m,\kappa(j)})_{j\in\mathbb{N}}=(e_{j})_{j\in\mathbb{N}}$ is an orthonormal basis of $H^{-m}(\mathbb{T}^{n})$, and, by the same clause with $k=\kappa(j)$, $\langle\Phi,e_{j}\rangle_{H^{-m}}=\langle\Phi,\zeta_{m,\kappa(j)}\rangle_{H^{-m}}=\rho_{\kappa(j)}^{m}\Phi(\kappa(j))$ for every $\Phi\in H^{-m}(\mathbb{T}^{n})$.

Claim 2. By the weights clause of the negative Sobolev lemma, $0<\rho_{k}^{m}\le1$. Hence $a_{j}=\rho_{k}^{m}\rho_{k}^{m}$ is positive, as a product of positive numbers, and multiplying $\rho_{k}^{m}\le1$ by the nonnegative number $\rho_{k}^{m}$ gives $a_{j}\le\rho_{k}^{m}\le1$. Thus $a$ is a sequence of positive real numbers with $a_{j}\le\bar{a}$ for every $j$, where $\bar{a}=1$; by the definition of weight sequences, $a$ is a weight sequence, and $a_{j}\le1$ for every $j$.

Claim 3. The negative Sobolev lemma holds for every natural number in place of its $m$; applying its weights clause with $m+1\in\mathbb{N}$ in place of $m$ gives $0<\rho_{k}^{m+1}$ and $(\rho_{k}^{m+1})^{2}=\frac{1}{\mu_{k}^{m+1}}$. Hence $c_{j}=(\rho_{k}^{m+1})^{2}$ is positive, as the square of a positive number, and $c_{j}=\frac{1}{\mu_{\kappa(j)}^{m+1}}$. By the summability clause of the summability lemma, applied with $s=m+1$ (admissible since $n\le m+1$, see the preliminary) and with the injective map $\kappa$, the series $\sum_{j=1}^{\infty}\frac{1}{\mu_{\kappa(j)}^{m+1}}$ converges. Its terms are the numbers $c_{j}$, so the series $\sum_{j=1}^{\infty}c_{j}$ is the same series and converges. By the definition of variance sequences, $c$ is a variance sequence.

Claim 4. By the weights clause of the negative Sobolev lemma, $\rho_{k}^{m+1}=\rho_{k}^{m}\rho_{k}$, and $\rho_{k}^{2}=\frac{1}{\mu_{k}}$ by the choice of $\rho_{k}$ in Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families. Rearranging the product,

$$c_{j}=(\rho_{k}^{m}\rho_{k})(\rho_{k}^{m}\rho_{k})=(\rho_{k}^{m})^{2}\rho_{k}^{2}=a_{j}\,\frac{1}{\mu_{k}},$$

and multiplying by $\mu_{k}$, which is positive by the preliminary, gives $\mu_{k}c_{j}=a_{j}\bigl(\mu_{k}\frac{1}{\mu_{k}}\bigr)=a_{j}$, that is $a_{j}=\mu_{\kappa(j)}c_{j}$. Since $1\le\mu_{k}$ by the preliminary and $c_{j}$ is positive by claim 3, multiplying $1\le\mu_{k}$ by $c_{j}$ gives $c_{j}\le\mu_{k}c_{j}=a_{j}$.

Claim 5. The data of The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis are available: $H^{-m}(\mathbb{T}^{n})$ is a real Hilbert space with orthonormal basis $(e_{j})_{j\in\mathbb{N}}$ by claim 1, and $a$ is a weight sequence by claim 2. For $\Phi\in H^{-m}(\mathbb{T}^{n})$ its coordinates are $\Phi_{j}=\langle\Phi,e_{j}\rangle_{H^{-m}}=\rho_{\kappa(j)}^{m}\Phi(\kappa(j))$ by claim 1. Since $a_{j}=(\rho_{k}^{m})^{2}$ is positive, $a_{j}^{-1}(\rho_{k}^{m})^{2}=1$, so for $\Phi,\Psi\in H^{-m}(\mathbb{T}^{n})$

$$a_{j}^{-1}\Phi_{j}\Psi_{j}=a_{j}^{-1}(\rho_{k}^{m})^{2}\,\Phi(k)\Psi(k)=\Phi(\kappa(j))\,\Psi(\kappa(j))\qquad\text{for every }j\in\mathbb{N}.\qquad(*)$$

By the definition of the noise space, $X^{a}$ is the set of those $\Phi\in H^{-m}(\mathbb{T}^{n})$ for which $\sum_{j=1}^{\infty}a_{j}^{-1}\Phi_{j}^{2}$ converges; by $(*)$ with $\Psi=\Phi$ this series has the same terms as $\sum_{j=1}^{\infty}\Phi(\kappa(j))^{2}$, so $\Phi\in H^{-m}(\mathbb{T}^{n})$ lies in $X^{a}$ if and only if $\sum_{j=1}^{\infty}\Phi(\kappa(j))^{2}$ converges. By the definition of the noise pairing and $(*)$, $\langle\Phi,\Psi\rangle_{a}=\sum_{j=1}^{\infty}\Phi(\kappa(j))\Psi(\kappa(j))$ for $\Phi,\Psi\in X^{a}$.

Let $U\in L^{2}(\mathbb{T}^{n})$. Then $\hat{U}\in H^{-m}(\mathbb{T}^{n})$ by the embedding clause of the negative Sobolev lemma, and the series $\sum_{j=1}^{\infty}\hat{U}(\kappa(j))^{2}$ converges by Parseval's identity applied with $U'=U$; hence $\hat{U}\in X^{a}$, so $U\mapsto\hat{U}$ maps $L^{2}(\mathbb{T}^{n})$ into $X^{a}$. For $U,U'\in L^{2}(\mathbb{T}^{n})$ the formula for the noise pairing and Parseval's identity give $\langle\hat{U},\hat{U}'\rangle_{a}=\sum_{j=1}^{\infty}\hat{U}(\kappa(j))\hat{U}'(\kappa(j))=\langle U,U'\rangle_{L^{2}}$. The map is injective by injectivity of the Fourier coefficients. It is onto $X^{a}$: let $\Phi\in X^{a}$. Then $\Phi\in H^{-m}(\mathbb{T}^{n})$, which by the definition of the Sobolev space is a set of coefficient families, and $\sum_{j=1}^{\infty}\Phi(\kappa(j))^{2}$ converges by the characterisation of $X^{a}$ above; so by the Riesz--Fischer clause there is $U\in L^{2}(\mathbb{T}^{n})$ with $\hat{U}=\Phi$. Hence $U\mapsto\hat{U}$ is a bijection from $L^{2}(\mathbb{T}^{n})$ onto $X^{a}$.

Claim 6. By the weights clause of the negative Sobolev lemma, $0<\rho_{k}^{m}$, so $\rho_{k}^{m}$ is a nonnegative real number whose square is $a_{j}$. By Existence and Uniqueness of the Nonnegative Square Root, $a_{j}$, which is positive by claim 2, has exactly one nonnegative square root, so $a_{j}^{1/2}=\rho_{k}^{m}$. Hence $a_{j}^{1/2}e_{j}=\rho_{k}^{m}\zeta_{m,k}$. The vector operations of $H^{-m}(\mathbb{T}^{n})$ are the pointwise operations on coefficient families by the definition of the Sobolev space, and by the basis clause of the negative Sobolev lemma $\zeta_{m,k}(k')$ equals $\frac{1}{\rho_{k}^{m}}$ if $k'=k$ and $0$ if $k'\ne k$. Therefore $(\rho_{k}^{m}\zeta_{m,k})(k')=\rho_{k}^{m}\frac{1}{\rho_{k}^{m}}=1$ if $k'=k$ and $(\rho_{k}^{m}\zeta_{m,k})(k')=\rho_{k}^{m}\cdot0=0$ if $k'\ne k$ (by claim 1 of Zero Products and Elementary Identities in a Field). By the trigonometric classes clause, $\hat{E}_{k}$ takes the same values at every $k'\in\mathbb{Z}^{n}$, so the two coefficient families coincide: $a_{j}^{1/2}e_{j}=\hat{E}_{\kappa(j)}$.