[Paper Review] Hermite-Padé approximation for certain systems of meromorphic functions

U. Fidalgo Prieto, G. López Lagomasino+1 more|arXiv (Cornell University)|Oct 25, 2013

Mathematical functions and polynomials9 references3 citations

TL;DR

This paper establishes the convergence and zero distribution of type I and type II Hermite-Padé approximants for systems of meromorphic functions constructed as rational modifications of Nikishin systems. By analyzing the asymptotic behavior of the approximants' polynomials, it proves that the zeros of the approximants accumulate on the supports of the underlying measures and are attracted to the poles of the rational functions, extending classical Markov-type theorems to multivariate meromorphic settings.

ABSTRACT

We study the convergence of sequences of type I and type II Hermite-Padé approximants for certain systems of meromorphic functions made up of rational modifications of Nikishin systems of functions.

Motivation & Objective

• To investigate the convergence and zero distribution of type I and type II Hermite-Padé approximants for systems of meromorphic functions.
• To extend classical Markov and Stieltjes theorems on Padé approximation to systems involving rational modifications of Nikishin systems.
• To analyze the asymptotic behavior of the approximants' polynomials under multi-indexed convergence regimes.
• To establish conditions under which the zeros of the approximants cluster on the supports of the measures and are attracted to the poles of the rational functions.

Proposed method

• The study employs type I and type II Hermite-Padé approximants defined via formal power series expansions and polynomial interpolation conditions in descending powers of z.
• It analyzes the asymptotic distribution of zeros of the approximant polynomials by leveraging the argument principle and convergence of ratios of polynomials on compact subsets of the complex plane.
• The proofs rely on constructing sequences of multi-indices and using contradiction arguments to rule out spurious zero clustering on intervals disjoint from the supports.
• Key components include the use of Cauchy transforms of measures, rational perturbations of Nikishin systems, and the construction of auxiliary polynomials to control zero locations.
• The analysis incorporates the use of the function space H(C\Δ) to ensure holomorphicity and convergence properties of the approximants.
• It applies known results on orthogonal polynomials and ratio asymptotics, particularly from Gonchar and Rakhmanov, to derive convergence in the limit.

Experimental results

Research questions

• RQ1 How do the zeros of type I and type II Hermite-Padé approximants behave asymptotically for systems of meromorphic functions?
• RQ2 What conditions ensure the convergence of Hermite-Padé approximants to the original meromorphic functions in the complex plane?
• RQ3 How do the poles of the rational functions in the system influence the distribution of the approximants' zeros?
• RQ4 Can the classical Markov-Stieltjes theorem for Padé approximants be extended to multivariate systems with rational perturbations?
• RQ5 What is the role of the multi-index convergence regime in determining the limiting behavior of the approximants?

Key findings

• The zeros of the type I approximant polynomials a_{n,m} are shown to accumulate on Δ_m and are attracted to the zeros of t_j with multiplicity equal to the order of the zero.
• For each j ∈ {0,…,m−1}, a zero ζ of t_j of multiplicity κ attracts exactly κ zeros of a_{n,m} as |n|→∞ along appropriate multi-index sequences.
• The ratio a_{n,m}/a_{n,j} converges uniformly on compact subsets of C\Δ_m to a meromorphic function whose poles and zeros correspond to the zeros of t_m and t_j respectively.
• The type II approximants converge to the original meromorphic function f = ŝ + r in the sense of formal Laurent series expansions, with error terms decaying as O(1/z^{n_j+1}).
• The proof establishes that no more than |n|−D zeros of the combination p_{n,0}t_0 + Σ p_{n,j}t_j ŝ_{1,j} can lie on Δ⊂R\Δ_1, contradicting assumptions of excessive clustering.
• The asymptotic behavior of the approximants is fully characterized by the interplay between the supports of the measures, the poles of the rational functions, and the structure of the multi-indices.