A stable consistent explicit spectral-grid heat step

Let $D$ be the finite-support sequences. By [F4], $D$ is dense in $X$; it is invariant under $U(t)$ because $U(t)$ multiplies coordinates and does not add support. The heat IVP and its finite-time bound are supplied by [F3].

Since $m=\nu +1$, ${\tau }_m=m^{-2}>0$ and ${\tau }_m\to 0$ as $\nu \to \mathrm{\infty }$. Let $P_m$ be truncation to coordinates $0,\dots ,m-1$ and $R_m$ inclusion. Their norms are at most $1$ by the coordinate supremum estimates and at least $1$ by testing $e_0$, so both have norm $1$ by [F7]. Also $P_m{R}_m=I_{G_m}$ and $\ker P_m$ is invariant under $U({\tau }_m)$ because $U$ is diagonal. The finite maximum $L_m:={\max }_{0\le j<m}\mathrm{\mid }1-{\tau }_m{j}^2\mathrm{\mid }$ bounds every diagonal multiplier, so $\mathrm{\parallel }{Q}_{m}g{\mathrm{\parallel }}_{\mathrm{\infty }}\le L_m\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $Q_m$ is bounded. Thus [F2] defines the common-space step $S_m$.

For $j<m$, $0\le 1-j^2\mathrm{/}{m}^2\le 1$; on coordinates $j\ge m$ the lift uses the multipliers $\exp (-j^2\mathrm{/}{m}^2)\in (0,1]$ from [F3]. Hence every coordinate multiplier of $S_m$ lies in $[0,1]$. Therefore $\mathrm{\parallel }{S}_{m}x{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}$ for every $x\in c_0$, and [F7] gives $\mathrm{\parallel }{S}_m{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le 1$. The bound $\mathrm{\parallel }{S}_m^{n}x{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }x{\mathrm{\parallel }}_{\mathrm{\infty }}$ holds at $n=0$; if it holds at $n$, applying the contraction estimate to $S_m^{n}x$ proves it at $n+1$. Induction [F8] gives it for every $n$, and then [F7] gives $\mathrm{\parallel }{S}_m^n{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le 1$. The family is stable on every finite time interval.

Fix $v\in D$, supported in coordinates $0,\dots ,N$. For $m>N$, direct evaluation and $0<\exp (-r{\tau }_m{j}^2)\le 1$ give ${\max }_{\begin{array}{c}r\in \mathbb{N}\\ r{\tau }_m\le T\end{array}}\frac{\mathrm{\parallel }(S_m-U({\tau }_m))U(r{\tau }_m)v{\mathrm{\parallel }}_{\mathrm{\infty }}}{{\tau }_m}\le \mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}{\max }_{0\le j\le N}\frac{\mathrm{\mid }1-{\tau }_m{j}^2-\exp (-{\tau }_m{j}^2)\mathrm{\mid }}{{\tau }_m}\mathrm{.}$ For $j=0$ the quotient is zero. For each fixed $1\le j\le N$, put $z={\tau }_m{j}^2$; [F5] gives $\frac{\mathrm{\mid }1-z-\exp (-z)\mathrm{\mid }}{{\tau }_m}=j^2\mid \frac{1-\exp (-z)}{z}-1\mid \to 0$, because the quotient tends to ${\exp }^{\mathrm{\prime }}(0)=1$ as $z\to 0$. The maximum of these finitely many quotients tends to zero: for any tolerance choose a threshold for each $j$ (or use threshold $1$ if there is no positive $j$), then take the largest threshold. Multiplication by the fixed $\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}$ preserves this limit. The bound is independent of $r$, so the family is consistent on $D$.

By [F6] and the explicit DC assumption [A1], this consistent stable family converges to the heat semigroup in the common supremum norm, uniformly over grid times in each bounded time interval.