Successive Schur-Riesz Analysis for Approximation

Many approximation methods enlarge a trial space by adjoining function blocks generated by different operators. Exact redundancy and strong cross-level interaction can make coefficients nonunique and render pairwise or diagonal-dominance tests needlessly pessimistic. For $V_m=\sum_{\ell\leq m}S_\ell(E_\ell)$ in a Hilbert space $\mathcal H$, we quotient coefficients representing the same function and control successive orthogonal innovations to obtain Riesz bounds independent of $m$. The setting includes factored operators $S_\ell=T_\ell\circ\cdots\circ T_1:E_\ell\to\mathcal H$, with compatible intermediate spaces, but the theorem allows arbitrary bounded $S_\ell$. The same constants control approximation, truncation, perturbation, and levelwise error. A block Schur complement identifies the intrinsic new dimension and gives the exact reduction in squared best-approximation error, leading to a constructive enrichment procedure. Nonstationary and lifted examples give positive intrinsic bounds where diagonal-dominance estimates are negative or labelled Gram matrices are singular; adaptive and recycled-subspace calculations illustrate the distinct roles of representation stability and application-specific utility.