arXiv subjects

Erwin Miña-Díaz

Publications and source records attributed to Erwin Miña-Díaz.

16 recordsLinked to original sources

arXiv2026

Higher-order asymptotics for the energy of greedy sequences on the unit circle

For the Riesz and logarithmic energies, we consider a greedy sequence $(a_n)_{n=0}^\infty$ of points on the unit circle $S^1$ constructed in such a way that for every integer $N\geq 2$, the energy of the configuration $(a_0,\ldots,a_{N-2},x)$ attains its optimal value (say $E_N$) at $x=a_{N-1}$. We derive an asymptotic expansion for $E_N$ in terms of certain bounded, oscillatory sequences $H_{N}$, $K_{N}$, and $R_{N}$ with a doubling periodicity property. In particular, we recover the results of \cite{LopMc1,LopWag} showing that after a proper translation and scaling of $E_N$, one is left with a sequence $T_N$ that is bounded and divergent. We show that the limit points of the sequence $T_N$ fill a closed interval. This follows from our asymptotic formulae and an analogous density result for the limit points of the sequences $H_{N}$, $K_{N}$, and $R_{N}$. We also give a new, simpler proof of density results obtained in \cite{LopMin} for the optimal values of the potential generated by a greedy sequence.

math.CA↗

arXiv2025

Norms of Chebyshev and Faber polynomials on curves with corners and cusps

We prove that the $n$th Chebyshev polynomial $T_{n}$ of a piecewise Dini-smooth Jordan curve $Γ$ satisfies \[ \lim_{n\to\infty}\frac{\|T_{n}\|_Γ}{\mathrm{cap}(Γ)^n}=1, \] where $\|\cdot\|_Γ$ is the supremum norm over $Γ$ and $\mathrm{cap}(Γ)$ its logarithmic capacity. This extends earlier results for smooth curves to curves with corner singularities, including cusps. The proof makes use of weighted Faber polynomials, which we analyze using a Fourier analytic representation of the standard Faber polynomials due to Pommerenke. We moreover obtain new asymptotic bounds for the norm of Faber polynomials which are sharp if, for instance, all corners have exterior angle greater than $π$.

math.CV↗

arXiv2025

Chebyshev polynomials on equipotential curves

For an analytic function $ϕ(z)$ with a Laurent expansion at $\infty$ of the form \begin{equation*} ϕ(z)=z+c_{0}+\frac{c_{1}}{z}+\frac{c_{2}}{z^{2}}+\cdots, \end{equation*} the Faber polynomial $F_n$ of degree $n$ associated to $ϕ$ is the polynomial part of the Laurent series at $\infty$ of $ϕ(z)^n$. We prove that the $n$th Chebyshev polynomial $T_{n,L_r}$ for the equipotential curve $L_r=\{z\in \mathbb{C}:|ϕ(z)|=r \}$ converges to $F_n$ as $r\to\infty$. The proof makes use of the fact that zero is the strongly unique best approximation to the monomial $z^n$ on the unit circle by polynomials of degree less than $n$.

math.CV↗

arXiv2024

Asymptotics of the optimal values of potentials generated by greedy energy sequences on the unit circle

For the Riesz and logarithmic potentials, we consider greedy energy sequences $(a_n)_{n=0}^\infty$ on the unit circle $S^1$, constructed in such a way that for every $n\geq 1$, the discrete potential generated by the first $n$ points $a_0,\ldots,a_{n-1}$ of the sequence attains its minimum value (say $U_n$) at $a_n$. We obtain asymptotic formulae that describe the behavior of $U_n$ as $n\to\infty$, in terms of certain bounded arithmetic functions with a doubling periodicity property. As previously shown in \cite{LopMc2}, after properly translating and scaling $U_n$, one obtains a new sequence $(F_n)$ that is bounded and divergent. We find the exact value of $\liminf F_n$ (the value of $\limsup F_n$ was already given in \cite{LopMc2}), and show that the interval $[\liminf F_n,\limsup F_n]$ comprises all the limit points of the sequence $(F_n)$.

math.CA↗

arXiv2024

Asymptotics of Bergman polynomials for domains with reflection-invariant corners

We study the asymptotic behavior of the Bergman orthogonal polynomials $(p_n)_{n=0}^{\infty}$ for a class of bounded simply connected domains $D$. The class is defined by the requirement that conformal maps $φ$ of $D$ onto the unit disk extend analytically across the boundary $L$ of $D$, and that $φ'$ has a finite number of zeros $z_1,\ldots, z_q$ on $L$. The boundary $L$ is then piecewise analytic with corners at the zeros of $φ'$. A result of Stylianopoulos implies that a Carleman-type strong asymptotic formula for $p_n$ holds on the exterior domain $\mathbb{C}\setminus\overline{D}$. We prove that the same formula remains valid across $L\setminus\{z_1,\ldots,z_q\}$ and on a maximal open subset of $D$. As a consequence, the only boundary points that attract zeros of $p_n$ are the corners. This is in stark contrast to the case when $φ$ fails to admit an analytic extension past $L$, since when this happens the zero counting measure of $p_n$ is known to approach the equilibrium measure for $L$ along suitable subsequences.

math.CV↗

arXiv2023

Orthogonal polynomials in weighted Bergman spaces

Let $w$ be a weight on the unit disk $\mathbb{D}$ having the form \[w(z)=|v(z)|^2\prod_{k=1}^s\left|\frac{z-a_k}{1-z\overline{a}_k}\right|^{m_k}\,,\quad m_k>-2,\ |a_k|<1,\] where $v$ is analytic and free of zeros in $\overline{\mathbb{D}}$, and let $(p_n)_{n=0}^\infty$ be the sequence of polynomials ($p_n$ of degree $n$) orthonormal over $\mathbb{D}$ with respect to $w$. We give an integral representation for $p_n$ from which it is in principle possible to derive its asymptotic behavior as $n\to\infty$ at every point $z$ of the complex plane, the asymptotic analysis of the integral being primarily dependent on the nature of the first singularities encountered by the function $v(z)^{-1}\prod_{k=1}^s(1-z\overline{a}_k)^{-1}$.

math.CA↗

arXiv2023

A representation for the reproducing kernel of a weighted Bergman space

For a weight function in the unit disk which is the modulus of a finite product of powers of Blaschke factors, we give a canonical representation for the reproducing kernel of the corresponding weighted Bergman space in terms of the values of the kernel and its derivatives at the origin. This yields a formula for the contractive zero divisor of a Bergman space corresponding to a finite zero set.

math.CV↗

arXiv2019

Nikishin systems on star-like sets: algebraic properties and weak asymptotics of the associated multiple orthogonal polynomials

Polynomials $Q_n(z)$, $n=0,1,\ldots,$ that are multi-orthogonal with respect to a Nikishin system of $p\geq 1$ compactly supported measures over the star-like set of $p+1$ rays $S_+:=\{z\in \mathbb{C}: z^{p+1}\geq 0 \}$ are investigated. We prove that the Nikishin system is normal, that the polynomials satisfy a three-term recurrence relation of order $p+1$ of the form $z Q_{n}(z)=Q_{n+1}(z)+a_{n}\,Q_{n-p}(z)$ with $a_n>0$ for all $n\geq p$, and that the nonzero roots of $Q_n$ are all simple and located in $S_+$. Under the assumption of regularity (in the sense of Stahl and Totik) of the measures generating the Nikishin system, we describe the asymptotic zero distribution and weak behavior of the polynomials $Q_n$ in terms of a vector equilibrium problem for logarithmic potentials. Under the same regularity assumptions, a theorem on the convergence of the Hermite-Padé approximants to the Nikishin system of Cauchy transforms is proven.

math.CA↗

arXiv2019

Asymptotics of polynomials orthogonal over circular multiply connected domains

Let $D$ be a domain obtained by removing, out of the unit disk $\{z:|z|<1\}$, finitely many mutually disjoint closed disks, and for each integer $n\geq 0$, let $P_n(z)=z^n+\cdots$ be the monic $n$th-degree polynomial satisfying the planar orthogonality condition $\int_D P_n(z)\overline{z^m}dxdy=0$, $0\leq m<n$. Under a certain assumption on the domain $D$, we establish asymptotic expansions and formulae that describe the behavior of $P_n(z)$ as $n\to\infty$ at every point $z$ of the complex plane. We also give an asymptotic expansion for the squared norm $\int_D|P_n|^2dxdy$.

math.CA↗

arXiv2019

Universality for conditional measures of the sine point process

The sine process is a rigid point process on the real line, which means that for almost all configurations $X$, the number of points in an interval $I = [-R,R]$ is determined by the points of $X$ outside of $I$. In addition, the points in $I$ are an orthogonal polynomial ensemble on $I$ with a weight function that is determined by the points in $X \setminus I$. We prove a universality result that in particular implies that the correlation kernel of the orthogonal polynomial ensemble tends to the sine kernel as the length $|I|=2R$ tends to infinity, thereby answering a question posed by A.I. Bufetov.

math.PR↗

arXiv2014

On the leading coefficient of polynomials orthogonal over domains with corners

Let $G$ be the interior domain of a piecewise analytic Jordan curve without cusps. Let $\{p_n\}_{n=0}^\infty$ be the sequence of polynomials that are orthonormal over $G$ with respect to the area measure, with each $p_n$ having leading coefficient $λ_n>0$. N. Stylianopoulos has recently proven that the asymptotic behavior of $λ_n$ as $n\to\infty$ is given by \[ \frac{n+1}π\frac{γ^{2n+2}}{ λ_n^{2}}=1-α_n, \] where $α_n=O(1/n)$ as $n\to\infty$ and $γ$ is the reciprocal of the logarithmic capacity of the boundary $\partial G$. In this paper, we prove that the $O(1/n)$ estimate for the error term $α_n$ is, in general, best possible, by exhibiting an example for which \[ \liminf_{n\to\infty}\,nα_n>0. \] The proof makes use of the Faber polynomials, about which a conjecture is formulated.

math.CA↗

arXiv2012

Asymptotics of Carleman polynomials for level curves of the inverse of a shifted Zhukovsky transformation

This paper complements the recent investigation of \cite{DM} on the asymptotic behavior of polynomials orthogonal over the interior of an analytic Jordan curve $L$. We study the specific case of $L=\{z= w-1 +(w-1)^{-1},\ |w|=R\}$, for some $R>2$, providing an example that exhibits the new features discovered in \cite{DM}, and for which the asymptotic behavior of the orthogonal polynomials is established over the entire domain of orthogonality. Surprisingly, this variation of the classical example of the ellipse turns out to be quite sophisticated. After properly normalizing the corresponding orthonormal polynomials $p_n$, $n=0,1,...$, and on certain critical subregion of the orthogonality domain, a subsequence $\{p_{n_k}\}$ converges if and only if $\log_{μ^4}(n_k)$ converges modulo 1 ($μ$ being an important quantity associated to $L$). As a consequence, the limiting points of the sequence $\{p_n\}$ form a one parameter family of functions, the parameter's range being the interval $[0,1)$. The polynomials $p_n$ are much influenced by a certain integrand function, the explained behavior being the result of this integrand having a nonisolated singularity that is a cluster point of poles. The nature of this singularity sparks purely from geometric considerations, as opposed to the more common situation where the critical singularities come from the orthogonality weight.

math.CV↗

arXiv2009

Asymptotic behavior and zero distribution of Carleman orthogonal polynomials

Let $L$ be an analytic Jordan curve and let $\{p_n(z)\}_{n=0}^\infty$ be the sequence of polynomials that are orthonormal with respect to the area measure over the interior of $L$. A well-known result of Carleman states that \label{eq12} \lim_{n\to\infty}\frac{p_n(z)}{\sqrt{(n+1)/π} [ϕ(z)]^{n}}= ϕ'(z) locally uniformly on certain open neighborhood of the closed exterior of $L$, where $ϕ$ is the canonical conformal map of the exterior of $L$ onto the exterior of the unit circle. In this paper we extend the validity of (\ref{eq12}) to a maximal open set, every boundary point of which is an accumulation point of the zeros of the $p_n$'s. Some consequences on the limiting distribution of the zeros are discussed, and the results are illustrated with two concrete examples and numerical computations.

math.CA↗

arXiv2007

An expansion for polynomials orthogonal over an analytic Jordan curve

We consider polynomials that are orthogonal over an analytic Jordan curve L with respect to a positive analytic weight, and show that each such polynomial of sufficiently large degree can be expanded in a series of certain integral transforms that converges uniformly in the whole complex plane. This expansion yields, in particular and simultaneously, Szego's classical strong asymptotic formula and a new integral representation for the polynomials inside L. We further exploit such a representation to derive finer asymptotic results for weights having finitely many singularities (all of algebraic type) on a thin neighborhood of the orthogonality curve. Our results are a generalization of those previously obtained in [7] for the case of L being the unit circle.

math.CA↗

arXiv2007

An asymptotic integral representation for Carleman orthogonal polynomials

In this paper we investigate the asymptotic behavior of polynomials that are orthonormal over the interior domain of an analytic Jordan curve L with respect to area measure. We prove that, inside L, these polynomials behave asymptotically like a sequence of certain integrals involving the canonical conformal map of the exterior of L onto the exterior of the unit circle and certain meromorphic kernel function defined in terms of a conformal map of the interior of L onto the unit disk. The error term in the integral representation is proven to decay geometrically and sufficiently fast, allowing us to obtain more precise asymptotic formulas for the polynomials under certain additional geometric considerations. These formulas yield, in turn, fine results on the location, limiting distribution and accumulation points of the zeros of the polynomials.

math.CA↗

arXiv2007

On the asymptotic behavior of Faber polynomials for domains with piecewise analytic boundary

For a function g(w) analytic and univalent in {w:1<|w|<\infty} with a simple pole at \infty and a continuous extension to {w:|w|\geq 1}, we consider the Faber polynomials F_n(z), n=0,1,2,..., associated to g(w) via their generating function g'(w)/(g(w)-z)=\sum_{n=0}^\infty F_n(z)w^{-(n+1)}. Assuming that g(w) maps the unit circle T onto a piecewise analytic curve L whose exterior domain has no outward-pointing cusps, and under an additional assumption concerning the "Lehman expansion" of g(w) about those points of T mapped onto corners of L, we obtain asymptotic formulas for F_n(z) that yield fine results on the location, limiting distribution and accumulation points of the zeros of the Faber polynomials. The asymptotic formulas are shown to hold uniformly and the exact rate of decay of the error terms involved is provided.

math.CA↗