Browsing by Author "Aptekarev, Alexander I."
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Item Convergence of ray sequences of Frobenius-Padé approximants(IOP, 2017) Aptekarev, Alexander I.; Bogolubsky, Alexey I.; Yattselev, Maxim L.; Mathematical Sciences, School of ScienceLet $\widehat\sigma$ be a Cauchy transform of a possibly complex-valued Borel measure $\sigma$ and $\{p_n\}$ a system of orthonormal polynomials with respect to a measure $\mu$, where $\operatorname{supp}(\mu)\cap\operatorname{supp}(\sigma)=\varnothing$. An $(m,n)$th Frobenius-Padé approximant to $\widehat\sigma$ is a rational function $P/Q$, ${\deg(P)\leq m}$, $\deg(Q)\leq n$, such that the first $m+n+1$ Fourier coefficients of the remainder function $Q\widehat\sigma-P$ vanish when the form is developed into a series with respect to the polynomials $p_n$. We investigate the convergence of the Frobenius-Padé approximants to $\widehat\sigma$ along ray sequences ${n/(n+m+1)\to c>0}$, $n-1\leq m$, when $\mu$ and $\sigma$ are supported on intervals of the real line and their Radon-Nikodym derivatives with respect to the arcsine distribution of the corresponding interval are holomorphic functions.Item Hermite-Padé Approximants for a Pair of Cauchy Transforms with Overlapping Symmetric Supports(Wiley, 2017-03) Aptekarev, Alexander I.; Van Assche, Walter; Yattselev, Maxim L.; Department of Mathematical Sciences, School of ScienceHermite-Padé approximants of type II are vectors of rational functions with a 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 transform of a third measure (Nikishin systems). In this work we consider the case where the supports form two overlapping intervals (in a symmetric way) and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap. We derive Szegő-type formulae for the asymptotics of the approximants, identify the convergence and divergence domains (the divergence domains appear for Angelesco systems but are not present for Nikishin systems), and show the presence of overinterpolation (a feature peculiar for Nikishin systems but not for Angelesco systems). Our analysis is based on a Riemann-Hilbert problem for multiple orthogonal polynomials (the common denominator).Item Jacobi matrices on trees generated by Angelesco systems: Asymptotics of coefficients and essential spectrum(EMS, 2021) Aptekarev, Alexander I.; Denisov, Sergey A.; Yattselev, Maxim L.; Mathematical Sciences, School of ScienceWe continue studying the connection between Jacobi matrices defined on a tree and multiple orthogonal polynomials (MOPs) that was recently discovered. In this paper, we consider Angelesco systems formed by two analytic weights and obtain asymptotics of the recurrence coefficients and strong asymptotics of MOPs along all directions (including the marginal ones). These results are then applied to show that the essential spectrum of the related Jacobi matrix is the union of intervals of orthogonality.Item On the parametrization of a certain algebraic curve of genus 2(Springer, 2015-11) Aptekarev, Alexander I.; Toulyakov, Dmitry N.; Yattselev, Maxim L.; Department of Mathematical Sciences, School of ScienceA parametrization of a certain algebraic curve of genus 2, given by a cubic equa-tion, is obtained. This curve appears in the study of Hermite-Pade´ approximants for a pair of functions with overlapping branch points on the real line. The suggested method of parametrization can be applied to other cubic curves as well.Item Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials(AMS, 2020) Aptekarev, Alexander I.; Denisov, Sergey A.; Yattselev, Maxim L.; Mathematical Sciences, School of ScienceWe consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Green’s functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on to a higher dimension. We illustrate the importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.Item Szeg\H{o}-type asymptotics for ray sequences of Frobenius-Pad\’e approximants(arXiv, 2016) Aptekarev, Alexander I.; Bogolubsky, Alexey I.; Yattselev, Maxim I.; Mathematical Sciences, School of ScienceLet $\widehat\sigma$ be a Cauchy transform of a possibly complex-valued Borel measure $\sigma$ and $\{p_n\}$ be a system of orthonormal polynomials with respect to a measure $\mu$, $\mathrm{supp}(\mu)\cap\mathrm{supp}(\sigma)=\varnothing$. An $(m,n)$-th Frobenius-Pad\'e approximant to $\widehat\sigma$ is a rational function $P/Q$, $\mathrm{deg}(P)\leq m$, $\mathrm{deg}(Q)\leq n$, such that the first $m+n+1$ Fourier coefficients of the linear form $Q\widehat\sigma-P$ vanish when the form is developed into a series with respect to the polynomials $p_n$. We investigate the convergence of the Frobenius-Pad\'e approximants to $\widehat\sigma$ along ray sequences $\frac n{n+m+1}\to c>0$, $n-1\leq m$, when $\mu$ and $\sigma$ are supported on intervals on the real line and their Radon-Nikodym derivatives with respect to the arcsine distribution of the respective interval are holomorphic functions.