Browsing by Author "Yattselev, Maxim L."
Now showing 1 - 10 of 17
Results Per Page
Sort Options
Item An asymptotic expansion for the expected number of real zeros of Kac-Geronimus polynomials(Rocky Mountain Mathematics Consortium, 2021) Aljubran, Hanan; Yattselev, Maxim L.; Mathematical Sciences, School of ScienceLet {φi(z;α)}i=0∞, corresponding to α∈(−1,1), be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number of real zeros, say 𝔼n(α), of random polynomials Pn(z):= ∑i=0nηiφi(z;α), where η0,…,ηn are i.i.d. standard Gaussian random variables. When α=0, φi(z;0)=zi and Pn(z) are called Kac polynomials. In this case it was shown by Wilkins that 𝔼n(0) admits an asymptotic expansion of the form 𝔼n(0)∼2πlog(n+1)+ ∑p=0∞Ap(n+1)−p (Kac himself obtained the leading term of this expansion). In this work we obtain a similar expansion of 𝔼(α) for α≠0. As it turns out, the leading term of the asymptotics in this case is (1∕π)log(n+1).Item An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC(Elsevier, 2019-01) Aljubran, Hanan; Yattselev, Maxim L.; Mathematical Sciences, School of ScienceLet {φi}∞i=0 be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure μ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say En(μ), of random polynomials Pn(z):=∑i=0nηiφi(z), where η0,…,ηn are i.i.d. standard Gaussian random variables. When μ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that En(|dξ|) admits an asymptotic expansion of the form En(|dξ|)∼2πlog(n+1)+∑p=0∞Ap(n+1)−p (Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where μ is absolutely continuous with respect to arclength measure and its Radon-Nikodym derivative extends to a holomorphic non-vanishing function in some neighborhood of the unit circle. In this case En(μ) admits an analogous expansion with coefficients the Ap depending on the measure μ for p≥1 (the leading order term and A0 remain the same).Item Asymptotics of Polynomials Orthogonal on a Cross with a Jacobi-Type Weight(Springer, 2020) Barhoumi, Ahmad; Yattselev, Maxim L.; Mathematical Sciences, School of ScienceWe investigate asymptotic behavior of polynomials Qn(z) satisfying non-Hermitian orthogonality relations ∫ΔskQn(s)ρ(s)ds=0,k∈{0,…,n−1}, where Δ:=[−a,a]∪[−ib,ib], a,b>0, and ρ(s) is a Jacobi-type weight.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 NUTTALL’S THEOREM WITH ANALYTIC WEIGHTS ON ALGEBRAIC S-CONTOURS(Elsevier, 2015-02) Yattselev, Maxim L.; Department of Mathematical Sciences, School of ScienceGiven a function f holomorphic at infinity, the nth diagonal Padé approximant to f, denoted by [n/n]f, is a rational function of type (n,n) that has the highest order of contact with f at infinity. Nuttall’s theorem provides an asymptotic formula for the error of approximation f−[n/n]f in the case where f is the Cauchy integral of a smooth density with respect to the arcsine distribution on [−1,1]. In this note, Nuttall’s theorem is extended to Cauchy integrals of analytic densities on the so-called algebraic S-contours (in the sense of Nuttall and Stahl).Item On LR2-best rational approximants to Markov functions on several intervals(Elsevier, 2022-06) Yattselev, Maxim L.; Mathematical Sciences, School of ScienceLet f(z)=∫(z−x)−1dμ(x), where μ is a Borel measure supported on several subintervals of (−1,1) with smooth Radon–Nikodym derivative. We study strong asymptotic behavior of the error of approximation (f−rn)(z), where rn(z) is the LR2-best rational approximant to f(z) on the unit circle with n poles inside the unit disk.Item On smooth perturbations of Chebyshëv polynomials and ∂¯ -Riemann–Hilbert method(Cambridge University Press, 2023) Yattselev, Maxim L.; Mathematical Sciences, School of ScienceItem 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.