We propose an algorithm for producing Hermite-Padé polynomials of type I for an arbitrary tuple of $m+1$ formal power series $[f_0,\dots,f_m]$, $m\geq1$, about $z=0$ ($f_j\in{\mathbb C}[[z]]$) under the assumption that the series have a certain (`general position') nondegeneracy property. This algorithm is a straightforward extension of the classical Viskovatov algorithm for construction of Padé polynomials (for $m=1$ our algorithm coincides with the Viskovatov algorithm).
The algorithm proposed here is based on a recurrence relation and has the feature that all the Hermite-Padé polynomials corresponding to the multiindices $(k,k,k,\dots,k,k)$, $(k+1,k,k,\dots,k,k)$, $(k+1,k+1,k,\dots,k,k),\dots$, $(k+1,k+1,k+1,\dots,k+1,k)$ are already known by the time the algorithm produces the Hermite-Padé polynomials corresponding to the multiindex $(k+1,k+1,k+1,\dots,k+1,k+1)$.
We show how the Hermite-Padé polynomials corresponding to different multiindices can be found via this algorithm by changing appropriately the initial conditions.
The algorithm can be parallelized in $m+1$ independent evaluations at each $n$th step.
We study the logarihtnmic asymptotic of multiple orthogonal polynomials arising in a mixed type Hermite-Padé approximation problem associated with the rational perturbation of a Nikishin system of functions. The formulas obtained allow to give exact estimates of the rate of convergence of the corresponding Hermite-Padé approximants.
We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of simultaneous rational interpolants with a bounded number of poles. The conditions are expressed in terms of intrinsic properties of the system of functions used to build the approximants. Exact rates of convergence for these denominators and the simultaneous rational approximants are provided.
A survey of direct and inverse type results for row sequences of Padé and Hermite-Padé approximation is given. A conjecture is posed on an inverse type result for type II Hermite-Padé approximation when it is known that the sequence of common denominators of the approximating vector rational functions has a limit. Some inverse type results are proved for the so called incomplete Padé approximants which may lead to the proof of the conjecture and the connection is discussed.
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.
We give a Montessus de Ballore type theorem for row sequences of Hermite-Padé approximations of vector valued analytic functions refining some results in this direction due to P.R. Graves-Morris and E.B. Saff. We do this introducing the notion of incomplete Padé approximation which contains, in particular, simultaneous Padé approximation and may be applied in the study of other systems of approximants as well.
We obtain strong and uniform asymptotics in every domain of the complex plane for the scaled polynomials $a (3nz)$, $b (3nz)$, and $c (3nz)$ where $a$, $b$, and $c$ are the type II Hermite-Padé approximants to the exponential function of respective degrees $2n+2$, $2n$ and $2n$, defined by $a (z)e^{-z}-b (z)=Ø(z^{3n+2})$ and $a (z)e^{z}-c (z)=Ø(z^{3n+2})$ as $z\to 0$. Our analysis relies on a characterization of these polynomials in terms of a $3\times 3$ matrix Riemann-Hilbert problem which, as a consequence of the famous Mahler relations, corresponds by a simple transformation to a similar R…
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.
We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite-Pade approximation to the exponential function, defined by p(z)e^{-z}+q(z)+r(z)e^{z} = O(z^{3n+2}) as z -> 0. These polynomials are characterized by a Riemann-Hilbert problem for a 3x3 matrix valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and fu…
We obtain extensions of the Poincaré and Perron theorems for higher order recurrence relations and apply them to obtain an inverse type theorem for row sequences of (type II) Hermite-Padé approximation of a vector of formal power series.
In this work we present new results on the convergence of diagonal sequences of certain mixed type Hermite-Padé approximants of a Nikishin system. The study is motivated by a mixed Hermite-Padé approximation scheme used in the construction of solutions of a Degasperis-Procesi peakon problem and germane to the analysis of the inverse spectral problem for the discrete cubic string.
We discuss a new approach to realization of the well-known Weierstrass's programme on efficient continuation of an analytic element corresponding to a~multivalued analytic function with finite number of branch points. Our approach is based on the use of Hermite--Padé polynomials.
In the paper, we propose two new conjectures about the convergence of Hermite Approximants of multivalued analytic functions of Laguerre class ${\mathscr L}$. The conjectures are based in part on the numerical experiments, made recently by the authors in [26] and [27].
We study the convergence of type I Hermite-Padé approximation for a class of meromorphic functions obtained by adding a vector of rational functions with real coefficients to a Nikishin system of functions.
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.
We consider row sequences of (type II) Hermite-Padé approximations with common denominator associated with a vector ${\bf f}$ of formal power expansions about the origin. In terms of the asymptotic behavior of the sequence of common denominators, we describe some analytic properties of ${\bf f}$ and restate some conjectures corresponding to questions once posed by A. A. Gonchar for row sequences of Padé approximants.
A new approach to the problem of the zero distribution of Hermite-Padé polynomials of type I for a pair of functions $f_1,f_2$ forming a Nikishin system is discussed. Unlike the traditional vector approach, we give an answer in terms of a scalar equilibrium problem with harmonic external field, which is posed on a two-sheeted Riemann surface.
We introduce and analyze some numerical results obtained by the authors experimentally. These experiments are related to the well known problem about the distribution of the zeros of Hermite--Padé polynomials for a collection of three functions $[f_0 \equiv 1,f_1,f_2]$. The numerical results refer to two cases: a pair of functions $f_1,f_2$ forms an Angelesco system and a pair of functions $f_1=f,f_2=f^2$ forms a (generalized) Nikishin system. The authors hope that the obtained numerical results will set up a new conjectures about the limiting distribution of the zeros of Hermite--Padé polynom…
In this paper are discussed the results of new numerical experiments on zero distribution of type I Hermite-Padé polynomials of order $n=200$ for three different collections of three functions $[1,f_1,f_2]$. These results are obtained by the authors numerically and do not match any of the theoretical results that were proven so far. We consider three simple cases of multivalued analytic functions $f_1$ and $f_2$, with separated pairs of branch points belonging to the real line. In the first case both functions have two logarithmic branch points, in the second case they both have branch points …
Let $[f_0,\dots,f_m]$ be a tuple of series in nonnegative powers of $1/z$, $f_j(\infty)\neq0$. It is supposed that the tuple is in "general position". We give a construction of type I and type II Hermite--Padé polynomials to the given tuple of degrees $\leq{n}$ and $\leq{mn}$ respectively and the corresponding $(m+1)$-multi-indexes with the following property. Let $M_1(z)$ and $M_2(z)$ be two $(m+1)\times(m+1)$ polynomial matrices, $M_1(z),M_2(z)\in\operatorname{GL}(m+1,\mathbb C[z])$, generated by type I and type II Hermite--Padé polynomials respectively. Then we have $M_1(z)M_2(z)\equiv I_{m…
The study of sequences of polynomials satisfying high order recurrence relations is connected with the asymptotic behavior of multiple orthogonal polynomials, the convergence properties of type II Hermite-Padé approximation, and eigenvalue distribution of banded Toeplitz matrices. We present some results for the case of recurrences with constant coefficients which match what is known for the Chebyshev polynomials of the first kind. In particular, under appropriate assumptions, we show that the sequence of polynomials satisfies multiple orthogonality relations with respect to a Nikishin system …
We discuss the relation between the linear Tschebyshev-Padé approximations to analytic function $f$ and the diagonal type I Hermite-Padé polynomials for the tuple of functions $[1,f_1,f_2]$ where the pair of functions $f_1,f_2$ forms certain Nikishin system. An approach is proposed of how to extend the seminal Stahl's Theory for Padé approximations for multivalued analytic functions to the Tschebyshev-Padé approximations. The approach is based on the relation between Tschebyshev-Padé approximations and Hermite-Padé polynomials and also on a connection of Hermite-Padé polynomials and multipoint…
We consider the problem of zero distribution of the first kind Hermite--Padé polynomials associated with a vector function $\vec f = (f_1, \dots, f_s)$ whose components $f_k$ are functions with a finite number of branch points in plane. We assume that branch sets of component functions are well enough separated (which constitute the Angelesco case). Under this condition we prove a theorem on limit zero distribution for such polynomials. The limit measures are defined in terms of a known vector equilibrium problem. Proof of the theorem is based on the methods developed by H.~Stahl, A.~A.~Goncha…
We prove ratio asymptotic for sequences of multiple orthogonal polynomials with respect to a Nikishin system of measures ${\mathcal{N}}(σ_1,...,σ_m)$ such that for each $k$, the support of $σ_k$ consists of an interval $\widetildeΔ_k$, on which $σ_k^{\prime} > 0$ almost everywhere, and a set without accumulation points in $\mathbb{R} \setminus \widetildeΔ_k$.
Hermite-Padé approximants of type II are vectors of rational functions with common denominator that interpolate a given vector of power series at infinity with maximal order. We are interested in the situation when the approximated vector is given by a pair of Cauchy transforms of smooth complex measures supported on the real line. The convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals (Angelesco systems) or two intervals that coincide under the condition that the ratio of the measures is a restriction of the Cauchy transfo…
The existence of the limit distribution of the zeros of Hermite-Padé polynomials of type II for a pair of functions forming a Nikishin system is proved using the scalar equilibrium problem posed on the two-sheeted Riemann surface. The relation of the results obtained here to some results of H. Stahl (1988) is discussed. Results of numerical experiments are presented. The results of the present paper and those obtained in the earlier paper of the second author [28], [32], [33] are shown to be in good accordance with both H. Stahl's results and with results of numerical experiments.
For an arbitrary tuple of $m+1$ germs of analytic functions at a fixed point, we introduce the so-called polynomial Hermite-Padé $m$-system (of order $n$, $n\in\mathbb N$), which consists of $m$ tuples of polynomials; these tuples, which are indexed by a natural number $k\in[1,\dots,m]$, are called the $k$th polynomials of the Hermite-Padé $m$-system. We study the weak asymptotics of the polynomials of the Hermite-Padé $m$-system constructed at the point $\infty$ from the tuple of germs $[1, f_{1,\infty},\dotsc$, $f_{m,\infty}]$ of the functions $1, f_1,\dots,f_m$ that are meromorphic on some …
We present a brief introduction to the theory of multiple orthogonal polynomials on the basis of known results for an important class of measures known as Nikishin systems. For type I and type II multiple orthogonal polynomials with respect to such systems of measures, we describe some of their most relevant properties regarding location and distribution of zeros as well as their weak and ratio asymptotic behavior.
Given a system of functions $\textup{\textbf{F}}=(F_1,\ldots,F_d),$ analytic on a neighborhood of some compact subset $E$ of the complex plane with simply connected complement, we define a sequence of vector rational functions with common denominator in terms of the orthogonal expansions of the components $F_k, k=1,\ldots,d,$ with respect to a sequence of orthonormal polynomials associated with a measure $μ$ whose support is contained in $E$. Such sequences of vector rational functions resemble row sequences of type II Hermite-Padé approximants. Under appropriate assumptions on $μ,$ we give ne…
In the paper, we discuss how it would be possible to succeed in Stahl's novel approach, 1987--1988, to explore Hermite--Padé polynomials based on Riemann surface properties. In particular, we explore the limit zero distribution of type I Hermite--Padé polynomials $Q_{n,0},Q_{n,1},Q_{n,2}$, $\operatorname{deg}{Q_{n,j}}\leq{n}$, for a collection of three analytic elements $[1,f_\infty,f^2_\infty]$. The element $f_\infty$ is an element of a function $f$ from the class $\mathbb C(z,w)$ where $w$ is supposed to be from the class $Z_{\pm1/2}([-1,1])$ of multivalued analytic functions generated by th…
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.
The paper presents some heuristic results about the distribution of zeros of Hermite-Pade polynomials of first kind for the case of three functions $1,f,f^2$, where $f$ has the form $f(z): = \prod\limits_ {j = 1 } ^3 (z-a_j) ^ {α_j} $, $α_j \in \mathbb C\setminus \mathbb Z $, $ \sum \limits_ {j = 1 } ^ 3 α_j = 0 $, $ f (\infty) = 1 $. Answers are given in terms related to the problem of extreme minimum capacity of a plane Nuttall condenser, two plates of which intersect ("hooked" for each other) in a five points: the branch points $ a_1, a_2, a_3 $ of function $ f $, at $ v_1 = v $, where $ v …
