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Register a new userZero distribution for Angelesco Hermite--Pade polynomials
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Authors
Evguenii A. Rakhmanov
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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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In this paper are discussed the results of new numerical experiments on zero distribution of type I Hermite-Pade 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 of second order, and finally, in the third case they both have branch points of third order. All three cases may be considered as representative of the asymptotic theory of Hermite-Pade polynomials. In the first two cases the numerical zero distribution of type I Hermite-Pade polynomials are similar to each other, despite the different kind of branching. But neither the logarithmic case, nor the square root case can be explained from the asymptotic point of view of the theory of type I Hermite-Pade polynomials. The numerical results of the current paper might be considered as a challenge for the community of all experts on Hermite-Pade polynomials theory.
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 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 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--Pade 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--Pade polynomials.
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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