Proof of The Cut-Off Shifted Operators of the Wick-Square Problem: the Hypotheses of the Convergence Theorem with Constants Independent of the Cutoff, Convergence on Bounded Test Data, and Compactness of the Triple

Each result cited below is universally quantified over the data in its own statement.

Elementary facts about real numbers (arithmetic and order, absolute values, squares and their monotonicity on nonnegative numbers, finite sums and sums over finite sets, limits of sequences and their order properties, the Archimedean property) are those of The Real Numbers: Standing Notation and Background and are used without further mention. Cube sums $S_{N}(a)=\sum_{k\in\Gamma_{N}}a(k)$, cube-summability and lattice sums $\sum_{k\in\mathbb{Z}^{n}}a(k)$ are those of Cube Sums of Families on the Integer Lattice.

Step 0 (Standing facts). By The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §triple, $H=H^{-3}(\mathbb{T}^{n})$ and $V=H^{-2}(\mathbb{T}^{n})$ with the inner products and norms of orders $-3$ and $-2$; by The Negative-Order Sobolev Spaces of the Torus §space these, and $H^{-1}=H^{-1}(\mathbb{T}^{n})$, are linear subspaces of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ under the pointwise operations, and $H^{-1}\subseteq V\subseteq H$ 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 §inclusion. Hence sums, differences and scalar multiples of elements of $H$ are computed mode by mode. We record the following facts, valid for every mode $k\in\mathbb{Z}^{n}$.

(F1) By Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §sobolev, used with $m=3$ and with $m=2$: for $z\in H$ the family $k\mapsto z(k)^{2}/\mu_{k}^{3}$ is cube-summable with lattice sum $|z|_{H}^{2}$, and for $y\in V$ the family $k\mapsto y(k)^{2}/\mu_{k}^{2}$ is cube-summable with lattice sum $|y|_{V}^{2}$.

(F2) By Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §units, used with $m=3$ and $m=2$, $|e_{k}|_{H}^{2}=\tfrac{1}{\mu_{k}^{3}}$ and $|e_{k}|_{V}^{2}=\tfrac{1}{\mu_{k}^{2}}$; by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §pairings, $\langle z,e_{k}\rangle_{H}=z(k)/\mu_{k}^{3}$ for $z\in H$.

(F3) By The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §domain, $D(A)=H^{-1}$ and, for $x\in D(A)$, $Ax\in H$ is the family $k\mapsto\mu_{k}x(k)$.

(F4) By The Wick-Square Problem on the Torus: Standing Notation §modes, $1\le\mu_{k}$; hence $0<\tfrac{1}{\mu_{k}^{j}}\le\tfrac{1}{\mu_{k}}\le1$ for every $j\in\mathbb{N}$.

(F5) By The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §root, $0<q_{k}$, and by The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §profile, $q_{k}\le\tfrac{\beta}{2\mu_{k}}$; thus $0<2q_{k}\le\tfrac{\beta}{\mu_{k}}$.

(F6) Let $N\in\mathbb{N}$. By The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §coordinates, for every $z\in\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ the family $\Pi_{N}z$ equals $z(k)$ at $k\in\Gamma_{N}$ and $0$ at $k\notin\Gamma_{N}$, and lies in $H^{-1}$, hence in $V$ and in $H$; consequently $\Pi_{N}(z-z')=\Pi_{N}z-\Pi_{N}z'$. By The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem with Noise, Control and Cost Cut Off to a Cube of Modes §data, $B^{N}_{q}x=\Pi_{N}(B_{q}x)$ for $x\in V$, where $(B_{q}x)(k)=2q_{k}x(k)$ by The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §riccati, and $G_{N}(p,p')=\sum_{k\in\Gamma_{N}}\langle p,e_{k}\rangle_{H}\langle p',e_{k}\rangle_{H}$.

(F7) $\Gamma_{N}$ is a nonempty finite set, $\Gamma_{N}\subseteq\Gamma_{M}$ for $N\le M$, by The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite and The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §nested; and $k\notin\Gamma_{N}$ means that some component $k_{i}$ fails $-N\le k_{i}\le N$.

Step 1 (Inner products of $H$ as lattice sums). Let $z,w\in H$. Then $z+w$ and $z-w$ lie in $H$, so by (F1) the families $k\mapsto(z(k)+w(k))^{2}/\mu_{k}^{3}$ and $k\mapsto(z(k)-w(k))^{2}/\mu_{k}^{3}$ are cube-summable with lattice sums $|z+w|_{H}^{2}$ and $|z-w|_{H}^{2}$. Since $(z(k)+w(k))^{2}-(z(k)-w(k))^{2}=4z(k)w(k)$, the family $k\mapsto z(k)w(k)/\mu_{k}^{3}$ is $\tfrac14$ times the first family plus $-\tfrac14$ times the second, so by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear it is cube-summable with lattice sum $\tfrac14(|z+w|_{H}^{2}-|z-w|_{H}^{2})$. Expanding the squared norms, which satisfy $|z\pm w|_{H}^{2}=\langle z\pm w,z\pm w\rangle_{H}$ by Real Inner Product Space §norm, by the symmetry, additivity and homogeneity of $\langle\cdot,\cdot\rangle_{H}$ (Real Inner Product Space §inner-product) gives $|z\pm w|_{H}^{2}=|z|_{H}^{2}\pm2\langle z,w\rangle_{H}+|w|_{H}^{2}$, so

$$\langle z,w\rangle_{H}=\sum_{k\in\mathbb{Z}^{n}}\frac{z(k)w(k)}{\mu_{k}^{3}}.$$

If moreover $z(k)=0$ for every $k\notin\Gamma_{N}$, the family vanishes off the nonempty finite set $\Gamma_{N}$, and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support gives $\langle z,w\rangle_{H}=\sum_{k\in\Gamma_{N}}z(k)w(k)/\mu_{k}^{3}$.

Step 2 (The projection is an $H$-contraction). Let $z\in H$. By (F6), $\Pi_{N}z\in H$ vanishes off $\Gamma_{N}$, so Step 1 (with both arguments $\Pi_{N}z$) gives $|\Pi_{N}z|_{H}^{2}=\sum_{k\in\Gamma_{N}}z(k)^{2}/\mu_{k}^{3}$. The family $k\mapsto z(k)^{2}/\mu_{k}^{3}$ is nonnegative and, by (F1), cube-summable with lattice sum $|z|_{H}^{2}$, so Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative (with $E=\Gamma_{N}$) gives $|\Pi_{N}z|_{H}^{2}\le|z|_{H}^{2}$, hence $|\Pi_{N}z|_{H}\le|z|_{H}$, both norms being nonnegative.

Step 3 (Clause 1: uniform hypotheses). Fix $N\in\mathbb{N}$.

The form $G_{N}$. Each map $p\mapsto\langle p,e_{k}\rangle_{H}$ is linear, so $G_{N}$, a sum over the finite set $\Gamma_{N}$ of products of two such maps, is symmetric, additive and homogeneous in its first argument. By the Cauchy-Schwarz inequality The Cauchy-Schwarz Inequality in a Real Inner Product Space, $|\langle p,e_{k}\rangle_{H}|\le|p|_{H}|e_{k}|_{H}$, so $|G_{N}(p,p')|\le C_{N}|p|_{H}|p'|_{H}$ with $C_{N}=\sum_{k\in\Gamma_{N}}|e_{k}|_{H}^{2}$. Hence $G_{N}\in\mathrm{Sym}(H)$ by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §form. For $p\in H$ put $\xi_{p}(k)=\langle p,e_{k}\rangle_{H}^{2}\ge0$. Then $G_{N}(p,p)=S_{N}(\xi_{p})\ge0=0_{\mathrm{Sym}}(p,p)$. By The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §gradient-form (and The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §noise) $\xi_{p}$ is cube-summable with lattice sum $G(p,p)$, and $G(p,p)\le I_{H}(p,p)=|p|_{H}^{2}$; by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative, $S_{N}(\xi_{p})\le G(p,p)$. Thus $0\le G_{N}(p,p)\le|p|_{H}^{2}$ for every $p\in H$, which is $0_{\mathrm{Sym}}\preceq G_{N}\preceq I_{H}$ by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §order.

The map $B^{N}_{q}$. It maps $V$ into $H$ by (F6). Monotonicity: for $x,y\in V$, by (F6) the vector $B^{N}_{q}x-B^{N}_{q}y=\Pi_{N}(B_{q}x-B_{q}y)$ is the family equal to $2q_{k}(x(k)-y(k))$ on $\Gamma_{N}$ and $0$ off $\Gamma_{N}$, and $x-y\in H$; Step 1 gives

$$\langle B^{N}_{q}x-B^{N}_{q}y,\,x-y\rangle_{H}=\sum_{k\in\Gamma_{N}}\frac{2q_{k}(x(k)-y(k))^{2}}{\mu_{k}^{3}}\ge0,$$

a finite sum of nonnegative numbers by (F5). $A$-monotonicity: for $x\in D(A)$, $B^{N}_{q}x$ equals $2q_{k}x(k)$ on $\Gamma_{N}$ and $0$ off it, and $(Ax)(k)=\mu_{k}x(k)$ by (F3); Step 1 gives $\langle B^{N}_{q}x,Ax\rangle_{H}=\sum_{k\in\Gamma_{N}}2q_{k}x(k)^{2}/\mu_{k}^{2}\ge0$. Boundedness on $V$-bounded sets: let $R>0$; since $B_{q}$ is a monotone nonlinearity by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §drift, Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple §bounded provides $\beta_{R}\in\mathbb{R}$ with $|B_{q}x|_{H}\le\beta_{R}$ whenever $x\in V$ and $|x|_{V}\le R$, and then $|B^{N}_{q}x|_{H}=|\Pi_{N}(B_{q}x)|_{H}\le|B_{q}x|_{H}\le\beta_{R}$ by Step 2. By Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple §nonlinearity, $B^{N}_{q}$ is a monotone nonlinearity for $(H,V,A)$.

The cost. For $x,y\in V$, $\Pi_{N}x,\Pi_{N}y\in V$ by (F6), so $|g(\Pi_{N}x)|\le C_{g}$, and, using (F6) and Step 2 with $z=x-y\in H$,

$$|g(\Pi_{N}x)-g(\Pi_{N}y)|\le\ell_{g}|\Pi_{N}x-\Pi_{N}y|_{H}=\ell_{g}|\Pi_{N}(x-y)|_{H}\le\ell_{g}|x-y|_{H}.$$

This proves clause 1.

Step 4 (Tail estimate). Let $N\in\mathbb{N}$, let $a:\mathbb{Z}^{n}\to\mathbb{R}$ be cube-summable, let $b:\mathbb{Z}^{n}\to\mathbb{R}$ be nonnegative and cube-summable, and suppose $|a(k)|\le b(k)$ for every $k\notin\Gamma_{N}$. We claim

$$\Bigl|\sum_{k\in\mathbb{Z}^{n}}a(k)-S_{N}(a)\Bigr|\le\sum_{k\in\mathbb{Z}^{n}}b(k)-S_{N}(b)\le\sum_{k\in\mathbb{Z}^{n}}b(k).$$

Let $M\in\mathbb{N}$ with $M\ge N$, so $\Gamma_{N}\subseteq\Gamma_{M}$ by (F7). If $\Gamma_{M}=\Gamma_{N}$ then $S_{M}(a)-S_{N}(a)=0=S_{M}(b)-S_{N}(b)$. Otherwise $D=\Gamma_{M}\setminus\Gamma_{N}$ is nonempty and finite, splitting the sum over $\Gamma_{M}$ gives $S_{M}(a)-S_{N}(a)=\sum_{k\in D}a(k)$ and likewise for $b$, and since every $k\in D$ lies outside $\Gamma_{N}$,

$$|S_{M}(a)-S_{N}(a)|\le\sum_{k\in D}|a(k)|\le\sum_{k\in D}b(k)=S_{M}(b)-S_{N}(b).$$

As $M\to\infty$, $S_{M}(a)$ and $S_{M}(b)$ converge to the lattice sums of $a$ and $b$ by Cube Sums of Families on the Integer Lattice §lattice-sum; passing to the limit in the inequality, valid for all $M\ge N$, gives the first claimed inequality, and the second holds because $S_{N}(b)\ge0$, a finite sum of nonnegative numbers.

Step 5 (The weights outside large cubes are large). Put $c_{0}=4\pi^{2}$. By The Number Pi §pi, $\pi=2x_{0}$ with $0<x_{0}$ by The Least Positive Zero of the Cosine §least-zero, so $0<\pi$ and $0<c_{0}$. Let $N\in\mathbb{N}$ and $k\notin\Gamma_{N}$. By (F7) some component satisfies $k_{i}<-N$ or $N<k_{i}$, and since $k_{i}$ and $N$ are integers, $k_{i}\le-(N+1)$ or $N+1\le k_{i}$; in both cases $(N+1)^{2}\le k_{i}^{2}\le\lVert k\rVert^{2}$, the square of the Euclidean norm being the sum of the squares of the components. Hence, as $1\le N+1$,

$$\mu_{k}=1+c_{0}\lVert k\rVert^{2}\ge1+c_{0}(N+1)^{2}>c_{0}(N+1)^{2}\ge c_{0}(N+1)>c_{0}N .$$

Claim: for every real number $T$ there is $K\in\mathbb{N}$ such that $T<\mu_{k}$ for every $N\in\mathbb{N}$ with $N\ge K$ and every $k\notin\Gamma_{N}$. Indeed, by the Archimedean property choose $K\in\mathbb{N}$ with $\tfrac{T}{c_{0}}<K$; then for $N\ge K$ and $k\notin\Gamma_{N}$, $\mu_{k}>c_{0}N\ge c_{0}K>T$.

Step 6 (The cut-off trace is a cube sum). Let $X\in\mathrm{Sym}(V)$ and $a_{X}(k)=X(e_{k},e_{k})$. By The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §noise, $a_{X}$ is cube-summable with lattice sum $\mathrm{Tr}_{f}X$. By Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them §trace, $\mathrm{Tr}_{f^{N}}X$ is the sum of the series $\sum_{j=1}^{\infty}X(f^{N}_{j},f^{N}_{j})$, which converges because $f^{N}$ is square-summable in $V$ (The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem with Noise, Control and Cost Cut Off to a Cube of Modes §data). The enumeration $\kappa$ is a bijection from $\mathbb{N}$ onto $\mathbb{Z}^{n}$ (The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §noise, Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families), so $J=\{j\in\mathbb{N}:\kappa(j)\in\Gamma_{N}\}$ is a nonempty finite set, mapped bijectively onto $\Gamma_{N}$ by $\kappa$; let $j^{*}$ be its largest element. For $j\notin J$, $f^{N}_{j}=0_{H}$ and $X(0_{H},0_{H})=0$ by bilinearity. Hence for every $m\ge j^{*}$ the $m$th partial sum equals $\sum_{j\in J}X(e_{\kappa(j)},e_{\kappa(j)})$, which, reindexing the finite sum along the bijection $\kappa:J\to\Gamma_{N}$, is $S_{N}(a_{X})$. The partial sums are eventually constant, so

$$\mathrm{Tr}_{f^{N}}X=S_{N}(a_{X}),\qquad \mathrm{Tr}_{f}X-\mathrm{Tr}_{f^{N}}X=\sum_{k\in\mathbb{Z}^{n}}a_{X}(k)-S_{N}(a_{X}).$$

Step 7 (The difference of the shifted operators). By The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem on the Sobolev Triple of Order Two §operator and The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem with Noise, Control and Cost Cut Off to a Cube of Modes §operator, $F^{\sharp}$ and $F^{\sharp}_{N}$ are second-order equation operators on $H$ relative to $(H,V,A)$, with $W=D(A)$; their $\delta$-shifts are those of Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts §shifted. Let $\delta>0$, $R>0$, $N\in\mathbb{N}$, and let $(x,r,p,Y)$ be a test datum bounded by $R$ in the sense of Convergence of Second-Order Equation Operators on a Hilbert Triple on Bounded Test Data §bounded-data: $x\in D(A)$, $r\in\mathbb{R}$, $p\in H$, $Y\in\mathrm{Sym}(H)$, $|x|_{V}\le R$, $|r|\le R$, $|p|_{H}\le R$, $\lVert Y\rVert\le R$. Let $\theta=1$ when treating the shifts $F^{\sharp,-}_{N,\delta}$ and $F^{\sharp,-}_{\delta}$, and $\theta=-1$ when treating $F^{\sharp,+}_{N,\delta}$ and $F^{\sharp,+}_{\delta}$, and put

$$\tilde r=r+\theta\delta h(x),\qquad\tilde p=p+\theta\delta Ax\in H,\qquad X=Y|_{V}+\theta\delta I_{V}\in\mathrm{Sym}(V),$$

so that $X(u,u')=Y(u,u')+\theta\delta\langle u,u'\rangle_{V}$ for $u,u'\in V$ and, by (F3), $\tilde p(k)=p(k)+\theta\delta\mu_{k}x(k)$. By the definition of the shifts, the quantity to be estimated is $F^{\sharp}_{N}(x,\tilde r,\tilde p,X)-F^{\sharp}(x,\tilde r,\tilde p,X)$. The terms $\gamma\tilde r$ cancel, and so do the terms $\langle Ax,\tilde p\rangle_{H}$, leaving

$$F^{\sharp}_{N}(x,\tilde r,\tilde p,X)-F^{\sharp}(x,\tilde r,\tilde p,X)=D_{1}-D_{2}-D_{3}+D_{4},$$

where

$$D_{1}=\tfrac{\nu}{2}\bigl(\mathrm{Tr}_{f}X-\mathrm{Tr}_{f^{N}}X\bigr),\quad D_{2}=\tfrac12\bigl(G(\tilde p,\tilde p)-G_{N}(\tilde p,\tilde p)\bigr),\quad D_{3}=\langle B_{q}x-B^{N}_{q}x,\tilde p\rangle_{H},\quad D_{4}=g(x)-g(\Pi_{N}x).$$

Hence the absolute value of the difference is at most $|D_{1}|+|D_{2}|+|D_{3}|+|D_{4}|$. Note $x\in V$ since $D(A)\subseteq V$, and $|x|_{V}^{2}\le R^{2}$, $|p|_{H}^{2}\le R^{2}$.

Step 8 (Estimates of the four terms). Keep the data of Step 7. Let $\omega(k)=\tfrac{1}{\mu_{k}^{2}}$; since $n\le3<4$ by The Wick-Square Problem on the Torus: Standing Notation §dimension, Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §convergent with $s=2$ shows $\omega$ is cube-summable, with lattice sum $\sigma_{\omega}$ say, and we put $\tau_{N}=\sigma_{\omega}-S_{N}(\omega)$.

The trace term. By Step 6, $D_{1}=\tfrac{\nu}{2}(\sum_{k}a_{X}(k)-S_{N}(a_{X}))$. For every $k$, by the restriction $Y|_{V}(e_{k},e_{k})=Y(e_{k},e_{k})$, Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §bound and (F2), (F4),

$$|a_{X}(k)|\le|Y(e_{k},e_{k})|+\delta|e_{k}|_{V}^{2}\le\lVert Y\rVert\,|e_{k}|_{H}^{2}+\frac{\delta}{\mu_{k}^{2}}=\frac{\lVert Y\rVert}{\mu_{k}^{3}}+\frac{\delta}{\mu_{k}^{2}}\le\frac{R+\delta}{\mu_{k}^{2}} .$$

Step 4 with $b=(R+\delta)\omega$, cube-summable with lattice sum $(R+\delta)\sigma_{\omega}$ by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear, gives $|D_{1}|\le\tfrac{\nu}{2}(R+\delta)\tau_{N}$.

For the other three terms suppose in addition that $\vartheta>0$ is a real number with $\tfrac{1}{\mu_{k}}\le\vartheta$ for every $k\notin\Gamma_{N}$. Then, for $k\notin\Gamma_{N}$ and $j,l\in\mathbb{N}$, (F4) gives $\tfrac{1}{\mu_{k}^{j+l}}=\tfrac{1}{\mu_{k}^{j}}\cdot\tfrac{1}{\mu_{k}^{l}}\le\tfrac{\vartheta}{\mu_{k}^{l}}$.

The gradient-form term. Let $\xi_{\tilde p}(k)=\langle\tilde p,e_{k}\rangle_{H}^{2}\ge0$. By The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §gradient-form, $\xi_{\tilde p}$ is cube-summable with lattice sum $G(\tilde p,\tilde p)$, and $G_{N}(\tilde p,\tilde p)=S_{N}(\xi_{\tilde p})$ by (F6), so $2D_{2}=\sum_{k}\xi_{\tilde p}(k)-S_{N}(\xi_{\tilde p})$. By (F2), $\xi_{\tilde p}(k)=\tilde p(k)^{2}/\mu_{k}^{6}$, and using $(s+t)^{2}\le2s^{2}+2t^{2}$, for $k\notin\Gamma_{N}$,

$$\xi_{\tilde p}(k)\le\frac{2p(k)^{2}}{\mu_{k}^{6}}+\frac{2\delta^{2}x(k)^{2}}{\mu_{k}^{4}}\le2\vartheta\Bigl(\frac{p(k)^{2}}{\mu_{k}^{3}}+\frac{\delta^{2}x(k)^{2}}{\mu_{k}^{2}}\Bigr)=:b_{2}(k).$$

By (F1) and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear, $b_{2}$ is nonnegative and cube-summable with lattice sum $2\vartheta(|p|_{H}^{2}+\delta^{2}|x|_{V}^{2})$. Step 4 gives $|D_{2}|\le\vartheta(|p|_{H}^{2}+\delta^{2}|x|_{V}^{2})\le\vartheta(1+\delta^{2})R^{2}$.

The drift term. Let $u=B_{q}x-B^{N}_{q}x\in H$; by (F6), $u(k)=0$ for $k\in\Gamma_{N}$ and $u(k)=2q_{k}x(k)$ for $k\notin\Gamma_{N}$. By Step 1 the family $\chi(k)=u(k)\tilde p(k)/\mu_{k}^{3}$ is cube-summable with lattice sum $D_{3}$, and $S_{N}(\chi)=0$. For $k\notin\Gamma_{N}$, by (F5),

$$|\chi(k)|\le\frac{2q_{k}|x(k)|\,|p(k)|}{\mu_{k}^{3}}+\frac{2q_{k}\delta x(k)^{2}}{\mu_{k}^{2}}\le\frac{\beta|x(k)|\,|p(k)|}{\mu_{k}^{4}}+\frac{\beta\delta x(k)^{2}}{\mu_{k}^{3}} .$$

From $0\le(\mu_{k}|x(k)|-|p(k)|)^{2}$ we get $2\mu_{k}|x(k)||p(k)|\le\mu_{k}^{2}x(k)^{2}+p(k)^{2}$, and dividing by $2\mu_{k}^{5}>0$, $\tfrac{|x(k)||p(k)|}{\mu_{k}^{4}}\le\tfrac12\bigl(\tfrac{x(k)^{2}}{\mu_{k}^{3}}+\tfrac{p(k)^{2}}{\mu_{k}^{5}}\bigr)$. Hence, for $k\notin\Gamma_{N}$,

$$|\chi(k)|\le\beta\vartheta\Bigl(\bigl(\tfrac12+\delta\bigr)\frac{x(k)^{2}}{\mu_{k}^{2}}+\frac12\,\frac{p(k)^{2}}{\mu_{k}^{3}}\Bigr)=:b_{3}(k),$$

and $b_{3}$ is nonnegative and cube-summable with lattice sum $\beta\vartheta((\tfrac12+\delta)|x|_{V}^{2}+\tfrac12|p|_{H}^{2})$ by (F1) and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear. Step 4 gives $|D_{3}|\le\beta\vartheta(1+\delta)R^{2}$.

The cost term. Let $v=x-\Pi_{N}x\in H$; by (F6), $v(k)=0$ for $k\in\Gamma_{N}$ and $v(k)=x(k)$ for $k\notin\Gamma_{N}$. By (F1) the family $\psi(k)=v(k)^{2}/\mu_{k}^{3}$ is cube-summable with lattice sum $|v|_{H}^{2}$, and $S_{N}(\psi)=0$. For $k\notin\Gamma_{N}$, $\psi(k)\le\vartheta x(k)^{2}/\mu_{k}^{2}$, a nonnegative cube-summable family with lattice sum $\vartheta|x|_{V}^{2}$ by (F1). Step 4 gives $|v|_{H}^{2}\le\vartheta R^{2}$. Since $x,\Pi_{N}x\in V$, the Lipschitz hypothesis on $g$ gives $|D_{4}|\le\ell_{g}|v|_{H}$.

Step 9 (Clause 2: convergence). Let $\delta>0$, $R>0$ and $\varepsilon>0$ be given; the choices are made in the following order. First put $\eta=\tfrac{\varepsilon}{4(\ell_{g}+1)}>0$, and

$$\vartheta=\min\Bigl\{\frac{\varepsilon}{4(1+\delta^{2})R^{2}},\ \frac{\varepsilon}{4\beta(1+\delta)R^{2}},\ \frac{\eta^{2}}{R^{2}}\Bigr\}>0 .$$

Second, by Step 5 with $T=\tfrac{1}{\vartheta}$, choose $K\in\mathbb{N}$ with $\tfrac{1}{\vartheta}<\mu_{k}$, hence $\tfrac{1}{\mu_{k}}<\vartheta$, whenever $N\ge K$ and $k\notin\Gamma_{N}$. Third, since $S_{N}(\omega)$ converges to $\sigma_{\omega}$ (Cube Sums of Families on the Integer Lattice §lattice-sum), choose $N_{1}\in\mathbb{N}$ with $|\sigma_{\omega}-S_{N}(\omega)|\le\tfrac{\varepsilon}{2\nu(R+\delta)}$ for every $N\ge N_{1}$. Finally put $N_{0}=\max\{K,N_{1}\}$.

Let $N\ge N_{0}$ and let $(x,r,p,Y)$ be a test datum bounded by $R$. For each $\theta\in\{1,-1\}$, Steps 7 and 8 apply with this $\vartheta$, and give $|D_{1}|\le\tfrac{\nu}{2}(R+\delta)\tau_{N}\le\tfrac{\varepsilon}{4}$ (as $\tau_{N}\le|\tau_{N}|$), $|D_{2}|\le\vartheta(1+\delta^{2})R^{2}\le\tfrac{\varepsilon}{4}$, $|D_{3}|\le\beta\vartheta(1+\delta)R^{2}\le\tfrac{\varepsilon}{4}$, and $|v|_{H}^{2}\le\vartheta R^{2}\le\eta^{2}$, so $|v|_{H}\le\eta$ (both being nonnegative) and $|D_{4}|\le\ell_{g}\eta\le\tfrac{\varepsilon}{4}$. By Step 7,

$$\bigl|F^{\sharp,\mp}_{N,\delta}(x,r,p,Y)-F^{\sharp,\mp}_{\delta}(x,r,p,Y)\bigr|\le\varepsilon$$

for both shifts. Since $\delta$, $R$, $\varepsilon$ were arbitrary, $(F^{\sharp}_{N})_{N\in\mathbb{N}}$ converges to $F^{\sharp}$ on bounded test data in the sense of Convergence of Second-Order Equation Operators on a Hilbert Triple on Bounded Test Data §convergence. This proves clause 2.

Step 10 (Clause 3: compactness). By The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian §triple, used with $s=2$ (so that $H^{-(s+1)}=H^{-3}$ and $H^{-s}=H^{-2}$) and with the enumeration $\kappa$ of The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §noise, the triple $(H,V,A)$ is the diagonal Hilbert triple determined by the orthonormal basis $(\zeta_{3,\kappa(j)})_{j\in\mathbb{N}}$ of $H$ and the weights $\lambda_{j}=\mu_{\kappa(j)}$. These weights tend to infinity: let $M\in\mathbb{R}$; apply Step 5 with $T=M$ to obtain $K\in\mathbb{N}$, and take $N=K$ in its conclusion: then $M<\mu_{k}$ for every $k\notin\Gamma_{K}$. The set $J_{K}=\{j\in\mathbb{N}:\kappa(j)\in\Gamma_{K}\}$ is nonempty and finite, since $\kappa$ is a bijection and $\Gamma_{K}$ is nonempty and finite (F7); let $j_{0}$ be one more than its largest element. For $j\ge j_{0}$ one has $j\notin J_{K}$, so $\kappa(j)\notin\Gamma_{K}$ and $M<\mu_{\kappa(j)}=\lambda_{j}$. Now let $(x_{j})_{j\in\mathbb{N}}$ be a sequence in $V$ that is bounded in $V$, that is, whose set of terms $\{x_{j}:j\in\mathbb{N}\}$ is bounded in $(V,d_{V})$ in the sense of Bounded Subset of a Metric Space: there are $y\in V$ and a real $\rho>0$ with $d_{V}(y,x_{j})\le\rho$ for all $j$. By Real Inner Product Space §distance, $d_{V}(y,x_{j})=|y-x_{j}|_{V}=|x_{j}-y|_{V}$, so $|x_{j}|_{V}\le|x_{j}-y|_{V}+|y|_{V}\le\rho+|y|_{V}=:R'$, a nonnegative real number, by the triangle inequality. By In a Diagonal Hilbert Triple whose Weights Tend to Infinity, Bounded Sequences of the Form Space Have Subsequences Converging in the Ambient Space §subsequence there are $j_{1}<j_{2}<\cdots$ and $x\in V$ with $|x|_{V}\le R'$ such that $(x_{j_{i}})_{i\in\mathbb{N}}$ converges to $x$ in $(H,d_{H})$. This subsequence converges in $H$, which proves clause 3.