arXiv subjects
Publications and source records attributed to Maxim L. Yattselev.
We study the connection between AAA (triple A) algorithm and multi-point Padé approximation when the approximated functions belong to either the Nevanlinna-Pick class or are Pólya frequency series.
We study strong asymptotics of orthogonal polynomials on an interval with varying weights, extending the framework of Totik's theorem on weighted orthogonal polynomials. Our results generalize this theorem in several directions. In particular, we allow the supports of the associated equilibrium measures to converge to a proper subinterval of the original interval of orthogonality or even collapse to a point. These extensions are motivated by applications to multiple orthogonality where such varying measures arise naturally.
We consider solutions of the sinh-Gordon Painlevé III equation \[ u_{xx} + \frac{1}{x} u_x = \sinh u \] that are real on $(0,\infty)$. They are parametrized by the monodromy parameter $p\in\overline{\mathbb{C}}$, $|p|>1$, and an additional real parameter $s^{\mathbb{R}}$ when $p=\infty$. Our previous joint work with A. Its described the asymptotic behavior of these solutions as $x\to\infty$. Here, we describe the transition as $x, p\to \infty$, $2\Im(p)=-s^{\mathbb R}$, between singular solutions ($|p|<\infty$) and smooth solutions ($p=\infty$). In short, if we parametrize $|p|^2 = 1 + e^{2\varkappa x}$, then the smooth exponential asymptotics of the solutions extends to the region $\varkappa>1$, with a change of the leading order term at $\varkappa=2$; at $\varkappa=1$ the exponential behavior transitions into an elliptic asymptotics, which holds for all $0<\varkappa<1$; as $\varkappa$ decays to zero, elliptic asymptotics degenerates into trigonometric one, which holds for all $p$ fixed.
Let $μ_1$ and $μ_2$ be two complex-valued Borel measures on the real line such that $\operatorname{supp} μ_1 =[α_1,β_1] < \operatorname{supp} μ_2 =[α_2,β_2]$ and ${\rm d}μ_i(x) = -ρ_i(x){\rm d}x/2π{\rm i}$, where $ρ_i(x)$ is the restriction to $[α_i,β_i]$ of a function non-vanishing and holomorphic in some neighborhood of $[α_i,β_i]$. Strong asymptotics of multiple orthogonal polynomials is considered as their multi-indices $(n_1,n_2)$ tend to infinity in both coordinates. The main goal of this work is to show that the error terms in the asymptotic formulae are uniform with respect to $\min\{n_1,n_2\}$.
Motivated by the simplest case of tt*-Toda equations, we study the large and small $x$ asymptotics for $x>0$ of real solutions of the sinh-Godron Painlevé III($D_6$) equation. These solutions are parametrized through the monodromy data of the corresponding Riemann-Hilbert problem. This unified approach provides connection formulae between the behavior at the origin and infinity of the considered solutions.
We describe the pole-free regions of the one-parameter family of special solutions of P$_\mathrm{II}$, the second Painlevé equation, constructed from the Airy functions. This is achieved by exploiting the connection between these solutions and the recurrence coefficients of orthogonal polynomials that appear in the analysis of the ensemble of random matrices corresponding to the cubic potential.
In this note we prove that \[ j!\,2^N \, \binom{N+j-1}{j} \, {}_2F_1\left(\begin{matrix}-j,-2j \\ -N-j+1 \end{matrix};-1\right) = \sum_{l=0}^N \binom{N}{l}\prod_{i=0}^{j-1}2(2i+1+l), \] where $ N $ and $ j $ are positive integers, which resolves a question posed by Ercolani, Lega, and Tippings.
We show that the one-parameter family of special solutions of P$_\mathrm{II}$, the second Painlevé equation, constructed from the Airy functions, as well as associated solutions of P$_\mathrm{XXXIV}$ and S$_\mathrm{II}$, can be expressed via the recurrence coefficients of orthogonal polynomials that appear in the analysis of the Hermitian random matrix ensemble with a cubic potential. Exploiting this connection we show that solutions of P$_\mathrm{II}$ that depend only on the first Airy function $ \mathrm{Ai} $ (but not on $ \mathrm{Bi} $) possess a scaling limit in the pole free region, which includes a disk around the origin whose radius grows with the parameter. We then use the scaling limit to show that these solutions are monotone in the parameter on the negative real axis.
We investigate asymptotic behavior of polynomials $ Q_n(z) $ satisfying non-Hermitian orthogonality relations $$ \int_Δs^kQ_n(s)ρ(s)ds =0, \quad k\in\{0,\ldots,n-1\}, $$ where $ Δ$ is a Chebotarëv (minimal capacity) contour connecting three non-collinear points and $ ρ(s) $ is a Jacobi-type weight including a possible power-type singularity at the Chebotarëv center of $ Δ$.
$\bar\partial$-extension of the matrix Riemann-Hilbert method is used to study asymptotics of the polynomials $P_n(z)$ satisfying orthogonality relations \[ \int_{-1}^1 x^lP_n(x)\frac{ρ(x)dx}{\sqrt{1-x^2}}=0, \quad l\in\{0,\ldots,n-1\}, \] where $ρ(x)$ is a positive $m$ times continuously differentiable function on $[-1,1]$, $m\geq3$.
Let $ f(z)=\int(z-x)^{-1}dμ(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-r_n)(z) $, where $ r_n(z) $ is the $ L_{\mathbb R}^2$-best rational approximant to $ f(z) $ on the unit circle with $ n $ poles inside the unit disk.
We investigate the phase diagram of the complex cubic unitary ensemble of random matrices with the potential $V(M)=-\frac{1}{3}M^3+tM$ where $t$ is a complex parameter. As proven in our previous paper, the whole phase space of the model, $t\in\mathbb C$, is partitioned into two phase regions, $O_{\mathsf{one-cut}}$ and $O_{\mathsf{two-cut}}$, such that in $O_{\mathsf{one-cut}}$ the equilibrium measure is supported by one Jordan arc (cut) and in $O_{\mathsf{two-cut}}$ by two cuts. The regions $O_{\mathsf{one-cut}}$ and $O_{\mathsf{two-cut}}$ are separated by critical curves, which can be calculated in terms of critical trajectories of an auxiliary quadratic differential. In our previous work the one-cut phase region was investigated in detail. In the present paper we investigate the two-cut region. We prove that in the two-cut region the endpoints of the cuts are analytic functions of the real and imaginary parts of the parameter $t$, but not of the parameter $t$ itself. We also obtain the semiclassical asymptotics of the orthogonal polynomials associated with the ensemble of random matrices and their recurrence coefficients. The proofs are based on the Riemann--Hilbert approach to semiclassical asymptotics of the orthogonal polynomials and the theory of $S$-curves and quadratic differentials.
We continue studying the connection between Jacobi matrices defined on a tree and multiple orthogonal polynomials (MOPs) that was discovered previously by the authors. 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.
Let $ \{φ_i(z;α)\}_{i=0}^\infty $, corresponding to $ α\in(-1,1) $, be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number of real zeros, say $ \mathbb E_n(α) $, of random polynomials \[ P_n(z) := \sum_{i=0}^nη_iφ_i(z;α), \] where $ η_0,\dots,η_n $ are i.i.d. standard Gaussian random variables. When $ α=0 $, $ φ_i(z;0)=z^i $ and $ P_n(z)$ are called Kac polynomials. In this case it was shown by Wilkins that $ \mathbb E_n(0)$ admits an asymptotic expansion of the form \[ \mathbb E_n(0) \sim \frac2π\log(n+1) + \sum_{p=0}^\infty A_p(n+1)^{-p} \] (Kac himself obtained the leading term of this expansion). In this work we obtain a similar expansion of $ \mathbb E(α) $ for $ α\neq 0 $. As it turns out, the leading term of the asymptotics in this case is $ (1/π)\log(n+1) $.
We study Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each subspace, we find generators and the generalized eigenfunctions written in terms of the orthogonal polynomials. The spectrum and its spectral type are studied for large classes of orthogonality measures.
We investigate asymptotic behavior of polynomials $ Q_n(z) $ satisfying non-Hermitian orthogonality relations $$ \int_Δs^kQ_n(s)ρ(s)\dd s =0, \quad k\in\{0,\ldots,n-1\}, $$ where $ Δ:= [-a,a]\cup [-\ic b,\ic b] $, $ a,b>0 $, and $ ρ(s) $ is a Jacobi-type weight.
Let $ \{φ_i\}_{i=0}^\infty $ 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 $ \mathbb E_n(μ) $, of random polynomials \[ P_n(z) := \sum_{i=0}^nη_iφ_i(z), \] where $ η_0,\dots,η_n $ 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 $ \mathbb E_n(|\mathrm dξ|) $ admits an asymptotic expansion of the form \[ \mathbb E_n(|\mathrm dξ|) \sim \frac2π\log(n+1) + \sum_{p=0}^\infty A_p(n+1)^{-p} \] (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 $ \mathbb E_n(μ) $ admits an analogous expansion with coefficients the $ A_p $ depending on the measure $ μ$ for $ p\geq 1 $ (the leading order term and $ A_0 $ remain the same).
We 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 $\mathbb{Z}_+$ to higher dimension. We illustrate importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.