AI 中文总结
研究连续复矩阵值函数在Frobenius范数下的有理极小极大逼近，通过推导相关准则和条件，建立其存在性、最优性特征，将连续统与边界逼近联系起来，并与对偶问题及数值方法关联，为验证和计算提供理论基础。
AI 中文摘要
本文研究连续复矩阵值函数在Frobenius范数下的有理极小极大逼近，其中所有逼近项共享一个公分母。这推广了经典标量有理逼近，应用于系统建模、微波设计和非线性特征值问题。首先证明了在自身稠密的点集上此类矩阵值逼近的存在性，扩展了Walsh的基础标量结果。接着通过推导原始/对偶矩阵值Kolmogorov准则和全局最优性的Ruttan型充分条件，建立了局部和全局极小极大逼近的特征。对于连续统上的解析函数，通过最大模原理将连续统极小极大逼近与边界上的逼近及有限边界样本联系起来。最后，对于离散逼近，将这些条件与对偶问题及基于对偶的数值方法m-d-Lawson联系起来。这些结果为验证和计算矩阵值有理极小极大逼近提供了理论基础。
英文摘要
In this paper, we study rational minimax approximation for continuous complex matrix-valued functions in the Frobenius norm, where all approximant entries share a common denominator. This generalizes classical scalar rational approximation, with applications in system modeling, microwave design, and nonlinear eigenvalue problems. We first prove the existence of such matrix-valued approximants on point sets dense in themselves, extending Walsh's foundational scalar result. Next, we establish characterizations of the local and global minimax approximants by deriving primal/dual matrix-valued Kolmogorov criteria and a Ruttan-type sufficient condition for global optimality. For analytic functions on a continuum, we link continuum minimax approximation to approximation on its boundary, and finite boundary samples via the maximum norm principle. We show that Ruttan's sufficient optimality condition provides a certificate under which a minimax approximant obtained from the boundary or from a discrete set of boundary nodes also solves the original continuum problem. Finally, for discrete approximation, we connect these conditions to a dual problem and the related dual-based numerical method m-d-Lawson: when the original minimax problem admits a solution, strong duality is equivalent to Ruttan's sufficient optimality condition, and, the optimality equations underlying the m-d-Lawson iteration coincide with Kolmogorov's dual criteria. These results provide a theoretical basis for certifying and computing matrix-valued rational minimax approximants.
Comments28 pages
