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Register a new userDirect and Inverse Results for Multipoint Hermite-Pade Approximants
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Guillermo L\\'opez Lagomasino
Publication date
2018
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English
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Given a system of functions f = (f1, . . . , fd) analytic on a neighborhood of some compact subset E of the complex plane, we give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of multipoint Hermite-Pade approximants. The exact rate of convergence of these denominators and of the approximants themselves is given in terms of the analytic properties of the system of functions. These results allow to detect the location of the poles of the system of functions which are in some sense closest to E.
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Given a vector function ${bf F}=(F_1,ldots,F_d),$ 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 expansions of the components $F_k, k=1,ldots,d,$ with respect to the sequence of Faber polynomials associated with $E$. Such sequences of vector rational functions are analogous to row sequences of type II Hermite-Pade approximation. We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of the sequence of vector rational functions so constructed. The exact rate of convergence of these denominators is provided and the rate of convergence of the approximants is estimated. It is shown that the common denominators of the approximants detect the poles of the system of functions closest to $E$ and their order.
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].
Given a regular compact set $E$ in the complex plane, a unit measure $mu$ supported by $partial E,$ a triangular point set $beta := {{beta_{n,k}}_{k=1}^n}_{n=1}^{infty},betasubset partial E$ and a function $f$, holomorphic on $E$, let $pi_{n,m}^{beta,f}$ be the associated multipoint $beta-$ Pade approximant of order $(n,m)$. We show that if the sequence $pi_{n,m}^{beta,f}, ninLambda, m-$ fixed, converges exact maximally to $f$, as $ntoinfty,ninLambda$ inside the maximal domain of $m-$ meromorphic continuability of $f$ relatively to the measure $mu,$ then the points $beta_{n,k}$ are uniformly distributed on $partial E$ with respect to the measure $mu$ as $ ninLambda$. Furthermore, a result about the zeros behavior of the exact maximally convergent sequence $Lambda$ is provided, under the condition that $Lambda$ is dense enough.
We propose an algorithm for producing Hermite-Pade polynomials of type I for an arbitrary tuple of $m+1$ formal power series $[f_0,dots,f_m]$, $mgeq1$, about $z=0$ ($f_jin{mathbb C}[[z]]$) 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 Pade 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-Pade 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-Pade polynomials corresponding to the multiindex $(k+1,k+1,k+1,dots,k+1,k+1)$. We show how the Hermite-Pade 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 consider the problem of zero distribution of the first kind Hermite--Pade 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.~Gonchar and the author. These methods obtained some further generalization in the paper in application to systems of polynomials defined by systems of complex orthogonality relations. Together with the characterization of the limit zero distributions of Hermite--Pade polynomials by a vector equilibrium problem we consider an alternative characterization using a Riemann surface $mathcal R(vec f)$ associated with $vec f$. In this terms we present a more general (without Angelesco condition) conjecture on the zero distribution of Hermite--Pade polynomials. Bibliography: 72 items.
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