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N. Levenberg

Publications and source records attributed to N. Levenberg.

At least 19 recordsLinked to original sources
arXiv2024

Random Sums of Weighted Orthogonal Polynomials in ${\mathbb C}^d$

We consider random polynomials of the form $G_n(z):= \sum_{|α|\leq n} ξ^{(n)}_αp_{n,α}(z)$ where $\{ξ^{(n)}_α\}_{|α|\leq n}$ are i.i.d. (complex) random variables and $\{p_{n,α}\}_{|α|\leq n}$ form a basis for $\mathcal P_n$, the holomorphic polynomials of degree at most $n$ in ${\mathbb C}^d$. In particular, this includes the setting where $\{p_{n,α}\}$ are orthonormal in a space $L^2(e^{-2n Q} τ)$, where $τ$ is a compactly supported Bernstein-Markov measure and $Q$ is a continuous weight function. Under an optimal moment condition on the random variables $\{ξ^{(n)}_α\}$, in dimension $d=1$ we prove convergence in probability of the zero measure to the weighted equilibrium measure, and in dimension $d \ge 2$ we prove convergence of zero currents.

math.PR↗
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↗
arXiv2022

Asymptotic Approximate Fekete Arrays

The notion of asymptotic Fekete arrays, arrays of points in a compact set $K\subset {\bf C}^d$ which behave asymptotically like Fekete arrays, has been well-studied, albeit much more recently in dimensions $d>1$. Here we show that one can allow a more flexible definition where the points in the array need not lie in $K$. Our results, which work in the general setting of weighted pluripotential theory, rely heavily, in the multidimensional setting, on the ground-breaking work of Berman, Boucksom and Nystrom.

math.CV↗
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

Optimal Polynomial Prediction Measures and Extremal Polynomial Growth

We show that the problem of finding the measure supported on a compact subset K of the complex plane such that the variance of the least squares predictor by polynomials of degree at most n at a point exterior to K is a minimum, is equivalent to the problem of finding the polynomial of degree at most n, bounded by 1 on K with extremal growth at this external point. We use this to find the polynomials of extremal growth for the interval [-1,1] at a purely imaginary point. The related problem on the extremal growth of real polynomials was studied by Erdős in 1947.

math.CA↗
arXiv2019

Pluripotential Theory and Convex Bodies: A Siciak-Zaharjuta theorem

We work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $({\bf R}^+)^d$. We define the {\it logarithmic indicator function} on ${\bf C}^d$: $$H_P(z):=\sup_{ J\in P} \log |z^{ J}|:=\sup_{ J\in P} \log[|z_1|^{ j_1}\cdots |z_d|^{ j_d}]$$ and an associated class of plurisubharmonic (psh) functions: $$L_P:=\{u\in PSH({\bf C}^d): u(z)- H_P(z) =0(1), \ |z| \to \infty \}.$$ We first show that $L_P$ is not closed under standard smoothing operations. However, utilizing a continuous regularization due to Ferrier which preserves $L_P$, we prove a general Siciak-Zaharjuta type-result in our $P-$setting: the weighted $P-$extremal function $$V_{P,K,Q}(z):=\sup \{u(z):u\in L_P, \ u\leq Q \ \hbox{on} \ K\}$$ associated to a compact set $K$ and an admissible weight $Q$ on $K$ can be obtained using the subclass of $L_P$ arising from functions of the form $\frac{1}{deg_P(p)}\log |p|$ (appropriately normalized).

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↗
arXiv2014

Extremal functions for real convex bodies

We study the smoothness of the Siciak-Zaharjuta extremal function associated to a convex body in $\mathbb{R}^2$. We also prove a formula relating the complex equilibrium measure of a convex body in $\mathbb{R}^n$ to that of its Robin indicatrix. The main tool we use are extremal ellipses.

math.CV↗
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

Polynomial Interpolation and Approximation in C^d

We update the state of the subject approximately 20 years after the publication of a previous article on this topic. This report is mostly a survey, with a sprinkling of assorted new results throughout.

math.CV↗
arXiv2011

Projective Hulls and Characterizations of Meromorphic Functions

We give conditions characterizing holomorphic and meromorphic functions in the unit disk of the complex plane in terms of certain weak forms of the maximum principle. Our work is directly inspired by recent results of John Wermer, and by the theory of the projective hull of a compact subset of complex projective space developed by Reese Harvey and Blaine Lawson.

math.CV↗