Modern applications [1] provide motivation to consider the tridiagonal Jacobi matrix (or the so-called discrete Schrödinger operator), a classical object of spectral theory, on graphs [2]. On homogeneous trees one method to implement such operators is based on Hermite–Padé interpolation problems (see [3]). Let μ⃗ = (μ1, . . . , μd) be a collection of positive Borel measures with compact supports on R. We denote by μ̂j(z) := ∫ (z−x)−1 dμj(x) their Cauchy transforms. For an arbitrary multi-index n⃗ ∈ Z+, we need to find polynomials qn⃗,0, qn⃗,1, . . . , qn⃗,d and pn⃗, pn⃗,1, . . . , pn⃗,d with deg pn⃗ = |n⃗| := n1 + · · · + nd such that the following interpolation conditions are satisfied as z →∞ for j = 1, . . . , d:

中文翻译：

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Nikishin 系统的多级插值和二叉树上 Jacobi 矩阵的有界性

现代应用 [1] 提供了在图 [2] 上考虑三对角雅可比矩阵（或所谓的离散薛定谔算子）的动机，它是谱理论的经典对象。在同构树上，实现此类算子的一种方法是基于 Hermite-Padé 插值问题（参见 [3]）。令 μ⃗ = (μ1, . . ., μd) 是在 R 上具有紧支撑的正 Borel 度量的集合。我们用 μ̂j(z) := ∫ (z−x)−1 dμj(x) 表示它们的柯西变换。对于任意多索引 n⃗ ∈ Z+，我们需要找到多项式 qn⃗,0, qn⃗,1, 。. . , qn⃗,d 和 pn⃗, pn⃗,1, . . . , pn⃗,d 度数 pn⃗ = |n⃗| := n1 + · · · + nd 使得满足以下插值条件为 z →∞ 对于 j = 1, . . . , d: