Multilevel interpolation for Nikishin systems and boundedness of Jacobi matrices on binary trees

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: