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 $V_m={\sum }_{\mathrm{\ell }\le m}{S}_{\mathrm{\ell }}(E_{\mathrm{\ell }})$ 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 $S_{\mathrm{\ell }}=T_{\mathrm{\ell }}\circ \cdots \circ T_1:E_{\mathrm{\ell }}\to \mathcal{H}$, with compatible intermediate spaces, but the theorem allows arbitrary bounded $S_{\mathrm{\ell }}$. 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.