Ralitza K. Kovacheva | Searcharxiv

arXiv2019

On overconvergent subsequencs of closed to rows classical Pade' approximants

Let $f$ be a power series with positive radius of convergence. In the present paper, we study the phenomenon of overconvergence of sequences of classical Pade' approximants pi{n,m_n} associated with f, where m(n)<=m(n+1)<=m(n) and m(n) = o(n/\log n), resp. m(n) = 0(n) as n is going to infiity. We extend classical results by J. Hadamard and A. A. Ostrowski related to overconvergent Taylor polynomials, as well as results by G. Lo'pez Lagomasino and A. Ferna'ndes Infante concerning overconvergent subsequences of a fixed row of the Pade' table.

math.CV↗

arXiv2016

Zero Distribution of Hermite-Padé Polynomials and Convergence Properties of Hermite Approximants for Multivalued Analytic Functions

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].

math.CV↗

arXiv2015

On the limit zero distribution of type I Hermite-Padé polynomials

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 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-Padé polynomials. In the first two cases the numerical zero distribution of type I Hermite-Padé 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-Padé polynomials. The numerical results of the current paper might be considered as a challenge for the community of all experts on Hermite-Padé polynomials theory.

math.CV↗