[Paper Review] Nikishin systems are perfect. Case of unbounded and touching supports

Fidalgo Prieto, Ulises+1 more|arXiv (Cornell University)|Sep 10, 2010

Quantum chaos and dynamical systems12 references3 citations

TL;DR

This paper establishes that Nikishin systems remain perfect even when the generating measures have unbounded supports or touching supports, extending previous results on bounded, non-intersecting supports. The key contribution is proving full-degree normality of type I and II approximants under these conditions, leading to a generalized version of Stieltjes' theorem in simultaneous Hermite-Padé approximation.

ABSTRACT

K. Mahler introduced the concept of perfect systems in the theory of simultaneous Hermite-Padé approximation of analytic functions. Recently, we proved that Nikishin systems, generated by measures with bounded support and non-intersecting consecutive supports contained on the real line, are perfect. Here, we prove that they are also perfect when the supports of the generating measures are unbounded or touch at one point. As an application, we give a version of Stieltjes' theorem in the context of simultaneous Hermite-Padé approximation.

Motivation & Objective

• To extend the theory of perfect systems to Nikishin systems with unbounded or touching supports.
• To establish that type I and type II approximants maintain full degree (normality) under these generalized support conditions.
• To generalize Stieltjes' theorem in the context of simultaneous Hermite-Padé approximation for such systems.
• To prove uniform convergence in Hausdorff content for the approximants, ensuring holomorphic limit functions.

Proposed method

• Adapting techniques from prior work on bounded, non-intersecting supports to handle unbounded and touching supports.
• Using permutation and reordering of multi-indices to reduce the problem to canonical forms.
• Applying the concept of mixed-type orthogonal polynomials and biorthogonal systems to analyze the structure of approximants.
• Leveraging the Hausdorff content convergence framework to establish uniform convergence on compact subsets.
• Employing the theory of Cauchy transforms and moment conditions to ensure existence and regularity of the measures.
• Extending results from [14] to include cases where supports are unbounded or intersect at a single point.

Experimental results

Research questions

• RQ1 Are Nikishin systems still perfect when the generating measures have unbounded supports?
• RQ2 Does normality of type I and II approximants persist when consecutive supports touch at a single point?
• RQ3 Can Stieltjes’ theorem be extended to the case of unbounded and touching supports in Nikishin systems?
• RQ4 What is the convergence behavior of Hermite-Padé approximants in the generalized setting of unbounded or touching supports?
• RQ5 How does the interlacing of zeros of consecutive linear forms behave under these generalized support conditions?

Key findings

• Nikishin systems remain perfect even when the supports of the generating measures are unbounded or touch at a single point.
• All multi-indices are normal for both type I and type II approximants, ensuring full-degree polynomials and uniqueness up to scalar multiples.
• The zeros of consecutive linear forms in the type II approximation interlace within the convex hull of the support of the first measure.
• Convergence of the approximants to the Cauchy transforms occurs uniformly on compact subsets of the extended complex plane minus the convex hull of the first measure’s support.
• The Hausdorff content convergence implies that the limit function is holomorphic, and the convergence is uniform on compacta.
• A generalized version of Stieltjes’ theorem holds for these extended Nikishin systems, confirming the asymptotic behavior of the approximants.