Finite-grid restriction, reconstruction, and common-space lift

Let $(X,U,D,({\tau }_{\nu }))$ be a common-space IVP interface from Common Banach-space interface for a linear IVP and its time schemes. A finite grid space $G_{\nu }$ is a normed vector space spanned by a possibly empty finite list over the same field $\mathbb{K}$. A grid realization consists, for each $\nu$, of bounded linear maps $P_{\nu }:X\to G_{\nu },R_{\nu }:G_{\nu }\to X,Q_{\nu }:G_{\nu }\to G_{\nu },$ where bounded linearity is in the sense of A bounded linear operator between normed spaces, such that $P_{\nu }{R}_{\nu }=I_{G_{\nu }}$ and $U({\tau }_{\nu })(\ker P_{\nu })\subseteq \ker P_{\nu },\ker P_{\nu }:=\{x\in X:P_{\nu }x=0\}\mathrm{.}$ Here $P_{\nu }$ is restriction to grid data, $R_{\nu }$ reconstructs grid data in $X$, and $Q_{\nu }$ is the one-step grid update.

The common-space lift of the grid update is $S_{\nu }:=R_{\nu }{Q}_{\nu }{P}_{\nu }+U({\tau }_{\nu })(I_X-R_{\nu }{P}_{\nu })\mathrm{.}$ The first summand advances the reconstructed grid component; the second continues the exact semigroup on the complementary component. The lift is bounded because all factors are bounded. More explicitly, if $c_P,c_R,c_Q$ bound $P_{\nu },R_{\nu },Q_{\nu }$ and $c_U$ bounds $U({\tau }_{\nu })$, then $\mathrm{\parallel }{S}_{\nu }x\mathrm{\parallel }\le (c_R{c}_Q{c}_P+c_U(1+c_R{c}_P))\mathrm{\parallel }x\mathrm{\parallel }$ by the factor bounds and the triangle inequality. This finite bound proves that $S_{\nu }$ is bounded. The invariance hypothesis on $\ker P_{\nu }$ is used later to identify the lift's powers when comparing grid and common-space stability. This definition is an analysis interface; it does not assert that evaluating the complementary semigroup is part of a finite-grid implementation.