arXiv subjects

F. Wielonsky

Publications and source records attributed to F. Wielonsky.

16 recordsLinked to original sources

arXiv2026

Coulomb equilibrium in the external field of an attractive-repellent pair of charges

The aim of this paper is to provide a complete analysis of the Coulomb equilibrium problem in the euclidean space $\mathbb{R}^d$, $d\geq2$, associated to the kernel $1/|x|^{d-2}$, with a non-convex external field created by an attractive-repellent pair of charges placed in $\mathbb{R}^{d+1} \setminus \mathbb{R}^d$. We consider the admissible setting, where the equilibrium measure is compactly supported, as well as the limiting weakly admissible setting, with a weaker external field at infinity, where the existence of the equilibrium measure still holds but possibly with an unbounded support. The main tools for our analysis are the notions of signed equilibrium and balayage of measures. We note that for certain configurations of charges and distances to the conductor, the support of the equilibrium measure is a shell (multidimensional annulus).

math.CA↗

arXiv2024

Weighted holomorphic polynomial approximation

For $G$ an open set in $\mathbb{C}$ and $W$ a non-vanishing holomorphic function in $G$, in the late 1990's, Pritsker and Varga characterized pairs $(G,W)$ having the property that any $f$ holomorphic in $G$ can be locally uniformly approximated in $G$ by weighted holomorphic polynomials $\{W(z)^np_n(z)\}, \ deg(p_n)\leq n$. We further develop their theory in first proving a quantitative Bernstein-Walsh type theorem for certain pairs $(G,W)$. Then we consider the special case where $W(z)=1/(1+z)$ and $G$ is a loop of the lemniscate $\{z\in \mathbb{C}: |z(z+1)|=1/4\}$. We show the normalized measures associated to the zeros of the $n-th$ order Taylor polynomial about $0$ of the function $(1+z)^{-n}$ converge to the weighted equilibrium measure of $\overline G$ with weight $|W|$ as $n\to \infty$. This mimics the motivational case of Pritsker and Varga where $G$ is the inside of the Szego curve and $W(z)=e^{-z}$. Lastly, we initiate a study of weighted holomorphic polynomial approximation in $\mathbb{C}^n, \ n>1$.

math.CV↗

arXiv2023

An extremal problem for the Bergman kernel of orthogonal polynomials

Let $Γ\subset \mathbb C$ be a curve of class $C(2,α)$. For $z_{0}$ in the unbounded component of ${\mathbb C}\setminus Γ$, and for $n=1,2,...$, let $ν_n$ be a probability measure with supp$(ν_{n})\subset Γ$ which minimizes the Bergman function $B_{n}(ν,z):=\sum_{k=0}^{n}|q_{k}^ν(z)|^{2}$ at $z_{0}$ among all probability measures $ν$ on $Γ$ (here, $\{q_{0}^ν,\ldots,q_{n}^ν\}$ are an orthonormal basis in $L^2(ν)$ for the holomorphic polynomials of degree at most $n$). We show that $\{ν_{n}\}_n$ tends weak-* to $\hatδ_{z_{0}}$, the balayage of the point mass at $z_0$ onto $Γ$, by relating this to an optimization problem for probability measures on the unit circle. Our proof makes use of estimates for Faber polynomials associated to $Γ$.

math.CV↗

arXiv2023

Boundary value problems and Heisenberg uniqueness pairs

We describe a general method for constructing Heisenberg uniqueness pairs $(Γ,Λ)$ in the euclidean space $\mathbb{R}^{n}$ based on the study of boundary value problems for partial differential equations. As a result, we show, for instance, that any pair made of the boundary $Γ$ of a bounded convex set $Ω$ and a sphere $Λ$ is an Heisenberg uniqueness pair if and only if the square of the radius of $Λ$ is not an eigenvalue of the Laplacian on $Ω$. The main ingredients for the proofs are the Paley-Wiener theorem, the uniqueness of a solution to a homogeneous Dirichlet or initial boundary value problem, the continuity of single layer potentials, and some complex analysis in $\mathbb{C}^{n}$. Denjoy's theorem on topological conjugacy of circle diffeomorphisms with irrational rotation numbers is also useful.

math.CA↗

arXiv2021

Polynomials associated to non-convex bodies

Polynomial spaces associated to a convex body $C$ in $({\bf R}^+)^d$ have been the object of recent studies. In this work, we consider polynomial spaces associated to non-convex $C$. We develop some basic pluripotential theory including notions of $C-$extremal plurisubharmonic functions $V_{C,K}$ for $K\subset {\bf C}^d$ compact. Using this, we discuss Bernstein-Walsh type polynomial approximation results and asymptotics of random polynomials in this non-convex setting.

math.CV↗

arXiv2020

C-transfinite diameter

We give a general formula for the $C-$transfinite diameter $δ_C(K)$ of a compact set $K\subset \mathbb{C}^2$ which is a product of univariate compacta where $C\subset (\mathbb{R}^+)^2$ is a convex body. Along the way we prove a Rumely type formula relating $δ_C(K)$ and the $C-$Robin function $ρ_{V_{C,K}}$ of the $C-$extremal plurisubharmonic function $V_{C,K}$ for $C \subset (\mathbb{R}^+)^2$ a triangle $T_{a,b}$ with vertices $(0,0), (b,0), (0,a)$. Finally, we show how the definition of $δ_C(K)$ can be extended to include many nonconvex bodies $C\subset \mathbb{R}^d$ for $d-$circled sets $K\subset \mathbb{C}^d$, and we prove an integral formula for $δ_C(K)$ which we use to compute a formula for the $C-$transfinite diameter of the Euclidean unit ball $\mathbb{B}\subset \mathbb{C}^2$.

math.CV↗

arXiv2019

Logarithmic potential theory and large deviation

We derive a general large deviation principle for a canonical sequence of probability measures, having its origins in random matrix theory, on unbounded sets $K$ of ${\bf C}$ with weakly admissible external fields $Q$ and very general measures $ν$ on $K$. For this we use logarithmic potential theory in ${\bf R}^{n}$, $n\geq 2$, and a standard contraction principle in large deviation theory which we apply from the two-dimensional sphere in ${\bf R}^{3}$ to the complex plane ${\bf C}$.

math.PR↗

arXiv2018

Zeros of Faber polynomials for Joukowski airfoils

Let $K$ be the closure of a bounded region in the complex plane with simply connected complement whose boundary is a piecewise analytic curve with at least one outward cusp. The asymptotics of zeros of Faber polynomials for $K$ are not understood in this general setting. Joukowski airfoils provide a particular class of such sets. We determine the (unique) weak-* limit of the full sequence of normalized counting measures of the Faber polynomials for Joukowski airfoils; it is never equal to the potential-theoretic equilibrium measure of $K$. This implies that many of these airfoils admit an electrostatic skeleton and also explains an interesting class of examples of Ullman related to Chebyshev quadrature.

math.CA↗

arXiv2015

Modified logarithmic potential theory and applications

We develop potential theory including a Bernstein-Walsh type estimate for functions of the form $p(z)q(f(z))$ where $p,q$ are polynomials and $f$ is holomorphic. Such functions arise in the study of certain ensembles of probability measures and our estimates lead to probabilistic results such as large deviation principles.

math.CA↗

arXiv2013

Vector Energy and Large Deviation

For d nonpolar compact sets K_1,...,K_d in the complex plane, d admissible weights Q_1,...,Q_d, and a positive semidefinite d x d interaction matrix C with no zero column, we define natural discretizations of the associated weighted vector energy of a d-tuple of positive measures μ=(μ_1,...,μ_d) where μ_j is supported in K_j and has mass r_j. We have an L^{\infty}-type discretization W(μ) and an L^2-type discretization J(μ) defined using a fixed measure ν=(ν_1,...,ν_d). This leads to a large deviation principle for a canonical sequence of probability measures on this space of d-tuples of positive measures if ν=(ν_1,...,ν_d) is a strong Bernstein-Markov measure.

math.CV↗

arXiv2011

On sequences of rational interpolants of the exponential function with unbounded interpolation points

We consider sequences of rational interpolants $r_n(z)$ of degree $n$ to the exponential function $e^z$ associated to a triangular scheme of complex points $\{z_{j}^{(2n)}\}_{j=0}^{2n}$, $n>0$, such that, for all $n$, $|z_{j}^{(2n)}|\leq cn^{1-α}$, $j=0,...,2n$, with $0<α\leq 1$ and $c>0$. We prove the local uniform convergence of $r_{n}(z)$ to $e^{z}$ in the complex plane, as $n$ tends to infinity, and show that the limit distributions of the conveniently scaled zeros and poles of $r_{n}$ are identical to the corresponding distributions of the classical Padé approximants. This extends previous results obtained in the case of bounded (or growing like $\log n$) interpolation points. To derive our results, we use the Deift-Zhou steepest descent method for Riemann-Hilbert problems. For interpolation points of order $n$, satisfying $|z_{j}^{(2n)}|\leq cn$, $c>0$, the above results are false if $c$ is large, e.g. $c\geq 2π$. In this connection, we display numerical experiments showing how the distributions of zeros and poles of the interpolants may be modified when considering different configurations of interpolation points with modulus of order $n$.

math.CA↗

arXiv2010

Non-intersecting squared Bessel paths: critical time and double scaling limit

We consider the double scaling limit for a model of $n$ non-intersecting squared Bessel processes in the confluent case: all paths start at time $t=0$ at the same positive value $x=a$, remain positive, and are conditioned to end at time $t=1$ at $x=0$. After appropriate rescaling, the paths fill a region in the $tx$--plane as $n\to \infty$ that intersects the hard edge at $x=0$ at a critical time $t=t^{*}$. In a previous paper (arXiv:0712.1333), the scaling limits for the positions of the paths at time $t\neq t^{*}$ were shown to be the usual scaling limits from random matrix theory. Here, we describe the limit as $n\to \infty$ of the correlation kernel at critical time $t^{*}$ and in the double scaling regime. We derive an integral representation for the limit kernel which bears some connections with the Pearcey kernel. The analysis is based on the study of a $3\times 3$ matrix valued Riemann-Hilbert problem by the Deift-Zhou steepest descent method. The main ingredient is the construction of a local parametrix at the origin, out of the solutions of a particular third-order linear differential equation, and its matching with a global parametrix.

math.CA↗

arXiv2007

Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights

We study a model of $n$ non-intersecting squared Bessel processes in the confluent case: all paths start at time $t = 0$ at the same positive value $x = a$, remain positive, and are conditioned to end at time $t = T$ at $x = 0$. In the limit $n \to \infty$, after appropriate rescaling, the paths fill out a region in the $tx$-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at $x = 0$, but at a certain critical time $t^*$ the smallest paths hit the hard edge and from then on are stuck to it. For $t \neq t^*$ we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time $t$ constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a $3 \times 3$ matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large $n$ limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.

math.CA↗

arXiv2005

Type II Hermite-Padé approximation to the exponential function

We obtain strong and uniform asymptotics in every domain of the complex plane for the scaled polynomials $a (3nz)$, $b (3nz)$, and $c (3nz)$ where $a$, $b$, and $c$ are the type II Hermite-Padé approximants to the exponential function of respective degrees $2n+2$, $2n$ and $2n$, defined by $a (z)e^{-z}-b (z)=Ø(z^{3n+2})$ and $a (z)e^{z}-c (z)=Ø(z^{3n+2})$ as $z\to 0$. Our analysis relies on a characterization of these polynomials in terms of a $3\times 3$ matrix Riemann-Hilbert problem which, as a consequence of the famous Mahler relations, corresponds by a simple transformation to a similar Riemann-Hilbert problem for type I Hermite-Padé approximants. Due to this relation, the study that was performed in previous work, based on the Deift-Zhou steepest descent method for Riemann-Hilbert problems, can be reused to establish our present results.

math.CA↗

arXiv2003

Asymptotic upper bounds for the entropy of orthogonal polynomials in the Szegő class

We give an asymptotic upper bound as $n\to\infty$ for the entropy integral $$E_n(w)= -\int p_n^2(x)\log (p_n^2(x))w(x)dx,$$ where $p_n$ is the $n$th degree orthonormal polynomial with respect to a weight $w(x)$ on $[-1,1]$ which belongs to the Szegő class. We also study two functionals closely related to the entropy integral. First, their asymptotic behavior is completely described for weights $w$ in the Bernstein class. Then, as for the entropy, we obtain asymptotic upper bounds for these two functionals when $w(x)$ belongs to the Szegő class. In each case, we give conditions for these upper bounds to be attained.

math.CA↗

arXiv2003

Quadratic Hermite-Pade approximation to the exponential function: a Riemann-Hilbert approach

We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite-Pade approximation to the exponential function, defined by p(z)e^{-z}+q(z)+r(z)e^{z} = O(z^{3n+2}) as z -> 0. These polynomials are characterized by a Riemann-Hilbert problem for a 3x3 matrix valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and functions derived from it. Our work complements recent results of Herbert Stahl.

math.CA↗