正双对角分解与混合型切比雪夫多正交多项式
Positive Bidiagonal Factorizations and Mixed-Type Chebyshev Multiple Orthogonal Polynomials
- Complutense University of Madrid（马德里康普顿斯大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文刻画了带状全正矩阵正双对角分解的存在性，通过Metelmann因子级数收敛性及Toeplitz符号负根给出全局分解，并推导混合型多正交多项式及其测度性质。
AI中文摘要：
我们刻画了带状全正矩阵何时允许正双对角分解。在有限维情形下，这样的分解总是存在的。对于具有任意有限带宽的半无限矩阵，其存在性等价于由Metelmann因子确定的有限族正级数的收敛性。它们的倒数正是标量Jacobi连分数的极限值。我们还证明了每个有界带状全正矩阵都可以在算子范数下被允许正双对角分解的矩阵逼近，且扰动被限制在固定的初始块内。对于带状Toeplitz矩阵，多项式符号的负根给出了具有有界因子的全局分解。这些根的有序划分确定了显式的混合型多正交多项式。应用现有的谱Favard定理，我们计算了它们的正测度矩阵：其支撑集是一个区间，且该测度是纯绝对连续的，具有秩一的代数密度。这些公式还给出了相关杀死随机游走的转移律和首返律。
英文摘要：
We characterize when a banded totally positive matrix admits a positive bidiagonal factorization. In finite dimension such a factorization always exists. For semi-infinite matrices of arbitrary finite bandwidth, existence is equivalent to convergence of a finite family of positive series determined by the Metelmann factors. Their reciprocals are the limiting values of scalar Jacobi continued fractions. We also prove that every bounded banded totally positive matrix can be approximated in operator norm by matrices admitting positive bidiagonal factorizations, through perturbations confined to a fixed initial block. For banded Toeplitz matrices, the negative roots of the polynomial symbol give a global factorization with bounded factors. Ordered partitions of these roots determine explicit mixed-type multiple orthogonal polynomials. Applying the existing spectral Favard theorem, we compute their matrix of positive measures: its support is one interval, and the measure is purely absolutely continuous with an algebraic density of rank one. These formulas also give transition and first-return laws for the associated killed random walks.
