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

Each result cited is universally quantified over the data in its own statement. Throughout, $f$ and $g$ are lattice fields, $t\in\mathbb{R}$, and $x_{j}=\langle x,e_{j}\rangle_{H^{-m}}$ for $x\in X$ and $j\in\mathbb{N}$. For a real number $r$ the square $r^{2}$ is the natural power, and $r^{2}=r\,r$ by that definition and the recursion of claim 1 of Properties of Finite Products, as $2=S(1)$ by Natural Numbers, as recorded in The Real Numbers: Standing Notation and Background §numbers (in force by A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §background); squares and products of a number with itself are used interchangeably below. The vector operations of $\mathrm{Map}(\mathbb{L}_{M},\mathbb{R})$ and of $X$ are the pointwise ones, by The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field §lattice-field and The Negative-Order Sobolev Spaces of the Torus §space. Write $T$ for the map $f\mapsto\hat{f}$ from $\mathrm{Map}(\mathbb{L}_{M},\mathbb{R})$ to $\mathrm{Map}(\Gamma_{M},\mathbb{R})$. The enumeration $\kappa:\mathbb{N}\to\mathbb{Z}^{n}$ is a bijection (Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families), and $\kappa^{-1}:\mathbb{Z}^{n}\to\mathbb{N}$ is its inverse.

Step 0: the numbers involved. By The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution, $L^{n/2}$ is the positive real number with $L^{n/2}L^{n/2}=L^{n}$ and $L^{-n/2}$ is its multiplicative inverse; hence $L^{-n/2}L^{n/2}=1$ and $L^{-n/2}L^{-n/2}=L^{-n}$. Let $j\in\mathbb{N}$. The number $a_{j}$ is positive, $a$ being a weight sequence by 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 §weights and Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space §weights. By A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §heads and Lifting Euclidean Tangent Fields of the Rescaled Head to Noise Tangent Fields on a Hilbert Space, $a_{j}^{1/2}$ is the positive square root of $a_{j}$ and $a_{j}^{-1/2}$ is the multiplicative inverse of $a_{j}^{1/2}$; by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root, $a_{j}^{1/2}$ is also the nonnegative square root of $a_{j}$ named in 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 §basis-vectors. Thus

$$a_{j}^{1/2}a_{j}^{1/2}=a_{j},\qquad a_{j}^{-1/2}a_{j}^{1/2}=1,\qquad a_{j}^{-1}a_{j}^{1/2}a_{j}^{1/2}=1 .$$

Step 1: reindexing over the active indices. Let $J_{M}$ be the set of 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 $N$: the set of those $j\in[d_{M}]$ with $\kappa(j)\in\Gamma_{M}$. For $k\in\Gamma_{M}$ the index $\kappa^{-1}(k)\in\mathbb{N}$ satisfies $\kappa(\kappa^{-1}(k))=k\in\Gamma_{M}$, so $\kappa^{-1}(k)\le d_{M}$, since $d_{M}$ is the largest $j\in\mathbb{N}$ with $\kappa(j)\in\Gamma_{M}$ by the same clause; hence $\theta(k)=\kappa^{-1}(k)$ defines a map $\theta:\Gamma_{M}\to J_{M}$. It is a bijection: for $j\in J_{M}$ and $k\in\Gamma_{M}$ one has $\theta(k)=j$ if and only if $k=\kappa(j)$, because $\kappa$ and $\kappa^{-1}$ are mutually inverse, and $\kappa(j)\in\Gamma_{M}$. In particular $J_{M}$ is nonempty, since $\Gamma_{M}$ is nonempty by The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field; and $J_{M}$ is finite, being a subset of $[d_{M}]$, which has $d_{M}$ elements by claim 1 of Basic Properties of Finite Sets and so is finite, by claim 3 of that lemma. Now let $y:[d_{M}]\to\mathbb{R}$ be a map with values $y_{j}$. The map $j\mapsto\chi_{M}(j)\,y_{j}$ on $[d_{M}]$ takes the value $0$ at every $j\in[d_{M}]$ with $j\notin J_{M}$ and the value $y_{j}$ at every $j\in J_{M}$. Hence, by claim 1 of Properties of a Sum over a Finite Index Set, by Sums over Finite Index Sets: Finite Unions, Disjoint Unions, Vanishing Terms, Dependent Pairs, Conjugation and the Modulus §vanishing, and by claim 2 of Properties of a Sum over a Finite Index Set applied to the bijection $\theta$,

$$\sum_{j=1}^{d_{M}}\chi_{M}(j)\,y_{j}=\sum_{j\in[d_{M}]}\chi_{M}(j)\,y_{j}=\sum_{j\in J_{M}}y_{j}=\sum_{k\in\Gamma_{M}}y_{\kappa^{-1}(k)}.\qquad\text{(R)}$$

Step 2: coordinates and the noise norm. Let $x\in X$ and $j\in\mathbb{N}$. By 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 §basis, applied with $\Phi=x$, and by 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 §basis-vectors,

$$x_{j}=\rho_{\kappa(j)}^{m}\,x(\kappa(j))=a_{j}^{1/2}\,x(\kappa(j)).\qquad\text{(C)}$$

By Step 0 it follows that $x(\kappa(j))=a_{j}^{-1/2}x_{j}$ and $a_{j}^{-1}x_{j}x_{j}=x(\kappa(j))^{2}$. The noise space and norm are those of The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis read as in 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, with the coordinates $x_{j}$. So the series $\sum_{j=1}^{\infty}a_{j}^{-1}x_{j}^{2}$ of The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §space is the series $\sum_{j=1}^{\infty}x(\kappa(j))^{2}$, both having the same terms: $x\in X^{a}$ if and only if this series converges, and then, by The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product and since $|x|_{a}$ is the nonnegative square root of $\langle x,x\rangle_{a}$,

$$|x|_{a}^{2}=\langle x,x\rangle_{a}=\sum_{j=1}^{\infty}x(\kappa(j))^{2}.\qquad\text{(N)}$$

Claim 2. Let $z\in\mathbb{L}_{M}$. By The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution §site-field and The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution §embedding, the values of $\mathcal{E}_{M}f$ on $\Gamma_{M}$ being $L^{n/2}\hat{f}(k)$, then by claim 4 of Properties of a Sum over a Finite Index Set with Step 0, and by Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum §inversion-formula,

$$\varphi_{\mathcal{E}_{M}f}(z)=L^{-n/2}\sum_{k\in\Gamma_{M}}L^{n/2}\hat{f}(k)\,\psi_{k}(z)=\sum_{k\in\Gamma_{M}}\hat{f}(k)\,\psi_{k}(z)=f(z).$$

So $\varphi_{\mathcal{E}_{M}f}=f$.

Claim 1. By Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum §bijection the map $T$ is linear, so $\widehat{f+g}(k)=\hat{f}(k)+\hat{g}(k)$ and $\widehat{tf}(k)=t\hat{f}(k)$ for $k\in\Gamma_{M}$. Hence, by The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution §embedding, for $k\in\Gamma_{M}$

$$\mathcal{E}_{M}(f+g)(k)=L^{n/2}\hat{f}(k)+L^{n/2}\hat{g}(k)=\mathcal{E}_{M}f(k)+\mathcal{E}_{M}g(k),\qquad\mathcal{E}_{M}(tf)(k)=t\,L^{n/2}\hat{f}(k)=t\,\mathcal{E}_{M}f(k),$$

while for $k\in\mathbb{Z}^{n}$ with $k\notin\Gamma_{M}$ all of $\mathcal{E}_{M}(f+g)(k)$, $\mathcal{E}_{M}(tf)(k)$, $\mathcal{E}_{M}f(k)$ and $\mathcal{E}_{M}g(k)$ are $0$, and $0=0+0=t\cdot0$. The operations of $X$ being pointwise, $\mathcal{E}_{M}(f+g)=\mathcal{E}_{M}f+\mathcal{E}_{M}g$ and $\mathcal{E}_{M}(tf)=t\,\mathcal{E}_{M}f$, so $\mathcal{E}_{M}$ is linear. If $\mathcal{E}_{M}f=\mathcal{E}_{M}g$, then by claim 2 $f=\varphi_{\mathcal{E}_{M}f}=\varphi_{\mathcal{E}_{M}g}=g$; so $\mathcal{E}_{M}$ is injective.

Claim 4. Let $x\in X$ and let $b\in\mathrm{Map}(\Gamma_{M},\mathbb{R})$ be given by $b(k)=L^{-n/2}x(k)$. By Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum §bijection, $T$ is a bijection whose inverse sends $b$ to the lattice field $z\mapsto\sum_{k\in\Gamma_{M}}L^{-n/2}x(k)\,\psi_{k}(z)$; by claim 4 of Properties of a Sum over a Finite Index Set and The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution §site-field this value equals $L^{-n/2}\sum_{k\in\Gamma_{M}}x(k)\,\psi_{k}(z)=\varphi_{x}(z)$. Hence $T^{-1}b=\varphi_{x}$, that is, $\hat{\varphi}_{x}=b$:

$$\hat{\varphi}_{x}(k)=L^{-n/2}x(k)\qquad(k\in\Gamma_{M}).\qquad\text{(F)}$$

By The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution §embedding and Step 0, $\mathcal{E}_{M}\varphi_{x}(k)=L^{n/2}L^{-n/2}x(k)=x(k)$ for $k\in\Gamma_{M}$, and $\mathcal{E}_{M}\varphi_{x}(k)=0$ for $k\in\mathbb{Z}^{n}$ with $k\notin\Gamma_{M}$.

Claim 7. The identity $x_{j}=a_{j}^{1/2}x(\kappa(j))$ is (C). Let $z\in\mathbb{L}_{M}$, and apply (R) with $y_{j}=a_{j}^{-1/2}x_{j}\,\psi_{\kappa(j)}(z)$, which equals $x(\kappa(j))\,\psi_{\kappa(j)}(z)$ by Step 2; then $y_{\kappa^{-1}(k)}=x(k)\,\psi_{k}(z)$ for $k\in\Gamma_{M}$, and (R) gives

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

Multiplying by $L^{-n/2}$ and using The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution §site-field gives the displayed formula of claim 7.

Claim 3. Let $z\in\mathbb{L}_{M}$. The space $X$ 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, in particular a real inner product space by Real Hilbert Space §hilbert. For $j\in[d_{M}]$ put $\beta_{j}=L^{-n/2}\chi_{M}(j)\,a_{j}^{-1/2}\,\psi_{\kappa(j)}(z)$, and let $v_{z}=\sum_{j=1}^{d_{M}}\beta_{j}e_{j}\in X$. By claim 7, by claim 3 of Properties of Finite Sums, and by the second identity of Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §combinations applied with $w=x$ and the $d_{M}$-tuple $(e_{1},\dots,e_{d_{M}})$, for every $x\in X$

$$\varphi_{x}(z)=\sum_{j=1}^{d_{M}}\beta_{j}\,x_{j}=\sum_{j=1}^{d_{M}}\beta_{j}\,\langle x,e_{j}\rangle_{H^{-m}}=\langle x,v_{z}\rangle_{H^{-m}}.$$

By Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces §functionals the map $x\mapsto\langle x,v_{z}\rangle_{H^{-m}}$ is a bounded linear functional on $X$. It is therefore a linear map from $X$ to $\mathbb{R}$ by Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces §real-line, and it is continuous from $(X,d_{H^{-m}})$ to $(\mathbb{R},d_{\mathbb{R}})$ by Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces §functionals, which extends Elementary Properties of Bounded Linear Maps and Functionals on Real Inner Product Spaces §lipschitz-continuous to bounded linear functionals.

Claim 5. Let $u=\mathcal{E}_{M}f$, an element of $X$ by The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution §embedding. For $j\in\mathbb{N}$ with $d_{M}<j$ one has $\kappa(j)\notin\Gamma_{M}$ by The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §head-dimension, so $u(\kappa(j))=0$; for $j\in[d_{M}]$ with $\chi_{M}(j)=0$ likewise $\kappa(j)\notin\Gamma_{M}$ and $u(\kappa(j))=0$, so that $u(\kappa(j))^{2}=\chi_{M}(j)\,u(\kappa(j))^{2}$ for every $j\in[d_{M}]$. Let $s_{N}=\sum_{j=1}^{N}u(\kappa(j))^{2}$ for $N\in\mathbb{N}$. Let $N\in\mathbb{N}$ with $d_{M}\le N$. Then $d_{M}<N+1$: one has $N<S(N)=N+1$ by Properties of the Order on the Natural Numbers §successor and Natural Numbers, so $d_{M}<N+1$ if $d_{M}=N$, and also if $d_{M}<N$, by the transitivity in Properties of the Order on the Natural Numbers §basic. Hence $\kappa(N+1)\notin\Gamma_{M}$ by The Galerkin Wick-Ordered Phi^4 Potential and Its Wick Constant on the Torus §head-dimension, read with cutoff $M$, so $u(\kappa(N+1))=0$ and the added term $u(\kappa(N+1))^{2}=0\cdot0$ is $0$; the recursion of claim 1 of Properties of Finite Sums therefore gives $s_{N+1}=s_{N}+u(\kappa(N+1))^{2}=s_{N}$, so by induction on $N$ one has $s_{N}=s_{d_{M}}$ for every $N\ge d_{M}$. Hence $|s_{N}-s_{d_{M}}|=0$ for $N\ge d_{M}$, and the sequence $(s_{N})_{N\in\mathbb{N}}$ converges to $s_{d_{M}}$ by Limit of a Sequence of Real Numbers, the index $d_{M}$ serving for every $\varepsilon>0$. By Series of Real Numbers §convergent the series $\sum_{j=1}^{\infty}u(\kappa(j))^{2}$ converges with sum $s_{d_{M}}$, so by Step 2 $u\in X^{a}$ and $|u|_{a}^{2}=s_{d_{M}}$. Finally, by the termwise identity above, by (R) with $y_{j}=u(\kappa(j))^{2}$, by The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution §embedding and Step 0, by claim 4 of Properties of a Sum over a Finite Index Set, and by Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum §parseval with $g=f$,

$$s_{d_{M}}=\sum_{j=1}^{d_{M}}\chi_{M}(j)\,u(\kappa(j))^{2}=\sum_{k\in\Gamma_{M}}u(k)^{2}=\sum_{k\in\Gamma_{M}}L^{n}\,\hat{f}(k)\,\hat{f}(k)=L^{n}\,L^{-n}\sum_{z\in\mathbb{L}_{M}}f(z)\,f(z)=\sum_{z\in\mathbb{L}_{M}}f(z)^{2}.$$

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

Claim 6. Let $h\in X^{a}$; then $h\in X$ by The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §space, so $\varphi_{h}$ is defined. By Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum §parseval with $f=g=\varphi_{h}$, by (F) and Step 0, and by claim 4 of Properties of a Sum over a Finite Index Set,

$$\sum_{z\in\mathbb{L}_{M}}\varphi_{h}(z)^{2}=L^{n}\Bigl(L^{-n}\sum_{z\in\mathbb{L}_{M}}\varphi_{h}(z)\,\varphi_{h}(z)\Bigr)=L^{n}\sum_{k\in\Gamma_{M}}\hat{\varphi}_{h}(k)\,\hat{\varphi}_{h}(k)=L^{n}\sum_{k\in\Gamma_{M}}L^{-n}\,h(k)^{2}=\sum_{k\in\Gamma_{M}}h(k)^{2},$$

which is the equality of claim 6. For the inequality, (R) with $y_{j}=h(\kappa(j))^{2}$ gives $\sum_{k\in\Gamma_{M}}h(k)^{2}=\sum_{j=1}^{d_{M}}\chi_{M}(j)\,h(\kappa(j))^{2}$. Since $0\le h(\kappa(j))^{2}$ and $\chi_{M}(j)\in\{0,1\}$, one has $\chi_{M}(j)\,h(\kappa(j))^{2}\le h(\kappa(j))^{2}$ for $j\in[d_{M}]$, so claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers gives

$$\sum_{k\in\Gamma_{M}}h(k)^{2}\le\sum_{j=1}^{d_{M}}h(\kappa(j))^{2}.$$

The right-hand side is the $d_{M}$-th partial sum of the series $\sum_{j=1}^{\infty}h(\kappa(j))^{2}$, whose terms are nonnegative and which converges with sum $|h|_{a}^{2}$ by Step 2, as $h\in X^{a}$. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates that partial sum is at most $|h|_{a}^{2}$, and therefore $\sum_{k\in\Gamma_{M}}h(k)^{2}\le|h|_{a}^{2}$.