全纯映射下的近最优柯尔莫哥洛夫宽度
Nearly optimal Kolmogorov widths under holomorphic mappings
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中文总结 AI 辅助
本文针对复巴拿赫空间间全纯映射下的柯尔莫哥洛夫宽度建立本质最优渐近界，回答了Cohen与DeVore的开放问题，构造了显式例子并应用于参数化偏微分方程解流形的可近似性刻画。
中文摘要 AI 辅助
本文针对复巴拿赫空间间全纯映射下的柯尔莫哥洛夫宽度，建立了本质最优的渐近界。给定一个柯尔莫哥洛夫宽度以速率s代数衰减的紧参数集，我们证明其在全纯映射下的像的宽度以任意速率t<s代数衰减，从而回答了Cohen与DeVore提出的开放问题。作为应用，我们得到了与inf-sup稳定参数化偏微分方程相关的解流形可近似性的精确刻画。我们还构造了一个显式例子，表明代数衰减指数的任意小损失是不可避免的。最后，我们对指数衰减 regime 下的柯尔莫哥洛夫宽度渐近界给出了类似刻画。我们的分析涉及巴拿赫空间中的多线性泰勒展开，以及一种新颖的块二进展开-截断技术。
英文摘要
This paper establishes essentially optimal asymptotic bounds for Kolmogorov widths under holomorphic mappings between complex Banach spaces. Given a compact parameter set whose Kolmogorov widths decay algebraically with rate s, we prove that the widths of its image under a holomorphic mapping decay algebraically with every rate t<s, thereby answering an open question raised by Cohen and DeVore. As an application, we obtain a sharp characterization of the approximability of solution manifolds associated with inf-sup stable parametrized PDEs. We also construct an explicit example showing that the arbitrarily small loss in the algebraic decay exponent is unavoidable. Finally, we provide similar characterization of asymptotic bounds for Kolmogorov widths in the exponentially decaying regime. Our analysis involves multilinear Taylor expansion in Banach spaces and a novel block dyadic expansion-truncation technique.
发表机构
- School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
