[Paper Review] On the convergence of multi-level Hermite-Padé approximants for a class of meromorphic functions

L. G. González Ricardo, G. López Lagomasino+1 more|arXiv (Cornell University)|Jan 22, 2020

Mathematical functions and polynomials14 references3 citations

TL;DR

This paper establishes the convergence of multi-level Hermite-Padé approximants for meromorphic functions formed by rational perturbations of Nikishin systems with real coefficients. It proves geometric convergence rates in compact subsets away from the supports and zeros of the characteristic polynomial, extending Markov and Stieltjes-type theorems to this class of functions using tools from potential theory and orthogonal polynomials.

ABSTRACT

The present paper deals with the convergence properties of multi-level Hermite-Padé approximants for a class of meromorphic functions given by rational perturbations with real coefficients of a Nikishin system of functions, and study the zero location of the corresponding multiple orthogonal polynomials.

Motivation & Objective

• To extend Markov and Stieltjes-type convergence theorems to multi-level Hermite-Padé approximants for meromorphic functions arising from rational perturbations of Nikishin systems.
• To analyze the zero distribution of multiple orthogonal polynomials associated with such perturbed systems.
• To establish uniform convergence in compact subsets of the extended complex plane away from the supports and poles of the approximants.
• To provide quantitative estimates on the rate of convergence using potential-theoretic tools and extremal problems.

Proposed method

• The study employs multi-level Hermite-Padé approximants defined via a system of polynomial equations involving Cauchy transforms of measures in a Nikishin system.
• It introduces a reversed Nikishin system to analyze the asymptotic behavior of the approximants and their associated polynomials.
• The convergence analysis relies on the use of the function $ \varphi_{\infty} $, which is the conformal map from the complement of the convex hull of the last interval to the exterior of the unit disk.
• It applies the maximum principle on curves $ \gamma_{\rho} $ with $ \rho < 1 $, using bounds on the norms of approximants and their ratios to derive decay estimates.
• The method uses the characteristic polynomial $ T(z) $ whose zeros determine the location of the poles of the approximants, and defines $ \delta(\mathcal{K}) $ as a modulus of continuity for the convergence rate.
• It leverages the fact that the ratio $ a_{n,j}/a_{n,m} $ converges uniformly to $ \widehat{s}_{m,j+1} $ on compact sets $ \mathcal{K} $, with the rate governed by $ \delta(\mathcal{K}) \|\varphi_{\infty}\|_{\mathcal{K}} $.

Experimental results

Research questions

• RQ1 Under what conditions do multi-level Hermite-Padé approximants converge for meromorphic functions obtained by rational perturbations of Nikishin systems?
• RQ2 How do the zeros of the associated multiple orthogonal polynomials behave asymptotically in the complex plane?
• RQ3 What is the precise rate of convergence of the approximants in compact subsets of the complex plane excluding the supports and poles?
• RQ4 Can geometric convergence rates be established for such approximants, and how do they depend on the geometry of the supports and the perturbation?
• RQ5 How does the reversed Nikishin system influence the convergence properties of the approximants?

Key findings

• The multi-level Hermite-Padé approximants converge uniformly on compact sets $ \mathcal{K} \subset \overline{\mathbb{C}} \setminus (\Delta_m \cup \{z: T(z) = 0\}) $, with the limit being the meromorphic function defined by the rational perturbation.
• The convergence rate is geometric, with the $ n $-th root of the error norm satisfying $ \limsup_{n \to \infty} \left\| \frac{a_{n,j}}{a_{n,m}} - \widehat{s}_{m,j+1} \right\|_{\mathcal{K}}^{1/n} \leq \delta(\mathcal{K}) \|\varphi_{\infty}\|_{\mathcal{K}} $.
• The zeros of the polynomial $ a_{n,m} $, which are the main determinants of the approximant's poles, converge to the zeros of the characteristic polynomial $ T(z) $ as $ n \to \infty $.
• For sufficiently large $ n $, the polynomial $ a_{n,m} $ has exactly $ D $ zeros outside $ \Delta_m $, and these remain uniformly bounded away from any compact set $ \mathcal{K} $ disjoint from $ \Delta_m $ and the zero set of $ T(z) $.
• The convergence of the approximants is uniform on $ \mathcal{K} $, and the error decays at a rate controlled by $ \delta(\mathcal{K}) $, a quantity related to the capacity of the set and the conformal map $ \varphi_{\infty} $.
• When the generating measures are regular in the sense of Stahl and Totik, more precise convergence estimates are expected, though these are left for future work.