On the convergence of type I Hermite–Padé approximants

Abstract

Padé approximation has two natural extensions to vector rational approximation through the so called type I and type II Hermite-Padé approximants. The convergence properties of type II Hermite-Padé approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite-Padé approximants for Nikishin systems of functions.

Asymptotics of Type I Hermite–Padé Polynomials for Semiclassical Functions

“…Remark 1.9. Similar results for Markov functions have been proved recently in [26,27,28]. For convenience, Theorems 1.6 and 1.8 assumed that |α| ∈ (0, 1/2), although the crucial constraint is that neither α nor 2α are integers.…”

Mixed type Hermite-Padé approximation inspired by the Degasperis-Procesi equation

“…We will make frequent use of [24,Theorem 1.3]. For convenience of the reader, we state it here as a lemma.…”

Asymptotic of Cauchy Biorthogonal Polynomials

“…We will make frequent use of [12,Theorem 1.3]. For convenience of the reader, we state it here as a lemma.…”