用于逼近的逐次Schur-Riesz分析
Successive Schur-Riesz Analysis for Approximation
浏览论文内容
中文总结 AI 辅助
该研究针对逼近方法中函数块冗余导致系数不唯一、检验结果悲观的问题，提出逐次Schur-Riesz分析方法，得到与层数无关的Riesz界，还给出构造性富集过程，验证了其在对角优势估计失效场景下的有效性。
中文摘要 AI 辅助
许多逼近方法通过邻接由不同算子生成的函数块来扩大试探空间。精确冗余性和强跨层相互作用可能导致系数不唯一，并使成对检验或对角优势检验得出不必要的悲观结论。针对希尔伯特空间$\mathcal{H}$中的$V_m=\sum_{\ell\leq m}S_\ell(E_\ell)$，我们对表示同一函数的系数取商，并控制逐次正交新息，以得到与$m$无关的Riesz界。该设定包含带相容中间空间的分解算子$S_\ell=T_\ell\circ\cdots\circ T_1:E_\ell\to\mathcal{H}$，但该定理允许任意有界$S_\ell$。相同的常数可控制逼近、截断、扰动和逐层误差。块Schur补可识别固有新维度，并给出最佳逼近误差平方的精确缩减量，由此得到一种构造性的富集过程。非平稳和提升算例在对角优势估计为负或标记Gram矩阵奇异的情况下给出了正的固有界；自适应和循环子空间计算阐明了表示稳定性与特定应用效用的不同作用。
英文摘要
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.
