Finite sampling reconstruction bridges compact-uniform raw quadrature limits

Let $X,U$ be real Banach spaces, and let continuous maps $H,H_n:X\to U$ satisfy ${\sup }_{a\in K}\mathrm{\parallel }{H}_n(a)-H(a){\mathrm{\parallel }}_U\to 0$ for every norm-compact $K\subset X$. Suppose continuous finite-dimensional sampling and reconstruction maps $S_n:X\to E_n$ and $J_n:E_n\to X$ are supplied and satisfy ${\sup }_{a\in K}\mathrm{\parallel }{J}_n{S}_n(a)-a{\mathrm{\parallel }}_X\to 0$ on every such $K$. Then $G_n(a)=H_n(J_n(S_n(a)))$ factors through finite-dimensional data, with $F_n=E_n$, finite map the identity and output reconstruction $R_n=H_n\circ J_n:E_n\to U$, and $G_n\to H$ uniformly on every norm-compact input set in the actual $U$ norm. When $X,U$ are function spaces on bounded domains with the pointwise output convention required by the discretization-invariant-family definition, this is a finite-sampling family in that definition.

For the supremum-norm continuous-input case these reconstruction hypotheses always hold with exactly the prescribed tags of arbitrary finite partitions of a nonempty bounded open $D\subset {\mathbb{R}}^d$ whose maximal cell diameters $h_n$ tend to zero. Namely let $X=C(\bar{D};{\mathbb{R}}^q)$, let $S_n(a)=(a({\xi }_{n,j}){)}_j$, and, when $h_n>0$, for $x\in \bar{D}$ put $w_{n,j}(x)=\max \{0,2h_n-\mathrm{\mid }x-{\xi }_{n,j}\mathrm{\mid }\},J_n(z)(x)=\frac{{\sum }_j{w}_{n,j}(x)z_j}{{\sum }_j{w}_{n,j}(x)}\mathrm{.}$ If $h_n=0$, the domain is the zero-dimensional singleton; define $J_n(z)$ to be the constant function with value $z_1$. This reconstructs every input exactly. In both branches $\mathrm{\parallel }{J}_n{S}_n{\mathrm{\parallel }}_{X\to X}\le 1$, and the reconstruction error is bounded by the input modulus of continuity at $2h_n$, uniformly on every compact input family. In a stronger Banach input norm the first clause applies whenever the stated reconstructions converge in that norm; convergence in the supremum norm alone is not substituted for that condition.

A further case needs no input reconstruction. Suppose a raw finite architecture first uses its input only through the sampled pointwise lift in the integral-only field $v_{1,n}(x)={\sum }_j{\omega }_{n,j}{\kappa }_n(x,{\xi }_{n,j})P(a({\xi }_{n,j}))+b_n(x)$, and every later stage acts on this field and the fixed parameters without further access to the original input. Assume these stages are defined and continuous on the compatible Banach spaces in question. Then each raw map itself factors through the finite input sample vector. Its compact-uniform raw convergence therefore already gives a finite-sampling family in the actual output norm, regardless of whether a sample reconstruction into the input norm has been supplied. This includes the integral-only universality subclass, when its kernels and intermediate spaces satisfy the declared compatible norm hypotheses.

Discretization-invariant operator family

Uniformly convergent images of a compact set have compact union

This statement is a local alteration and application of the compact-union mechanism in Kovachki et al., Appendix B, Lemma21, and its use for compact-uniform composition in the proof of Lemma22. The complete relevant source statements and printed proofs on PDF/printed pages74–77 of the cached JMLR full text were inspected in this provenance clarification. The source does not state the present finite-sampling bridge, the explicit normalized tag weights, or the integral-first factorization as these separate assertions; those adaptations are fully proved above. The source's printed closedness argument in Lemma21 assumes unjustified convergence of the selected integer indices. The locally proved open-cover compact-union supplier supplies the correct stated conclusion without relying on that step. The finite sampling/reconstruction interpretation is the explicit local definition of discretization invariance. No new retrieval, source-fetch stamp, whole-paper reread, or source attribution for the newly authored formulas is claimed.