Weighted polynomials and weighted pluripotential theory

Random polynomials and pluripotential-theoretic extremal functions

There is a natural pluripotential-theoretic extremal function V_{K,Q} associated to a closed subset K of C^m and a real-valued, continuous function Q on K. We define random polynomials H_n whose coefficients with respect to a related orthonormal basis are independent, identically distributed complex-valued random variables having a very general distribution (which includes both normalized complex and real Gaussian distributions) and we prove results on a.s. convergence of a sequence 1/n log |H_n| pointwise and in L^1_{loc}(C^m) to V_{K,Q}. In addition we obtain results on a.s. convergence of a…

Pluripotential Theory and Convex Bodies: Large Deviation Principle

We continue the study in a previous work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $({\bf R}^+)^d$. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms of $P-$pluripotential-theoretic notions. As an important preliminary step, we first give an existence proof for the solution of a Monge-Ampère equation in an appropriate finite energy class. This is achieved using a variational approach.

Sampling of real multivariate polynomials and pluripotential theory

We consider the problem of stable sampling of multivariate real polynomials of large degree in a general framework where the polynomials are defined on an affine real algebraic variety $M$, equipped with a weighted measure. In particular, this framework contains the well-known setting of trigonometric polynomials (when $M$ is a torus equipped with its invariant measure), where the limit of large degree corresponds to a high frequency limit, as well as the classical setting of one-variable orthogonal algebraic polynomials (when $M$ is the real line equipped with a suitable measure), where the s…

多重位势理论与凸体：一个Siciak-Zaharjuta定理

Pluripotential Theory and Convex Bodies: A Siciak-Zaharjuta theorem

本文在凸体P关联的多项式加权多重位势理论中，利用保持函数类L_P的Ferrier连续正则化方法，证明了加权P-极值函数可由特定规范化对数多项式函数子类给出的Siciak-Zaharjuta型定理。

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.

Statistical Mechanics of Interpolation Nodes, Pluripotential theory and Complex Geometry

This is mainly a survey, explaining how the probabilistic (statistical mechanical) construction of Kahler-Einstein metrics on compact complex manifolds, introduced in a series of works by the author, naturally arises from classical approximation and interpolation problems in complex n-space. A fair amount of background material is included. Along the way, the results are generalized to the non-compact setting of complex n-space. This yields a probabilistic construction of Kahler solutions to Einstein's equations in complex n-space, with cosmological constant -beta, from a gas of interpolation …

关于二变量八元数 Monge-Ampère 方程和与八元数 plurisubharmonic 函数相关的 pluripotential 理论

On octonionic Monge-Ampère equation and pluripotential theory associated to octonionic plurisubharmonic functions of two variables

本文研究了二变量八元数 plurisubharmonic 函数的 Monge-Ampère 方程和 pluripotential 理论，解决了八元数非结合性问题，并建立了相关函数的正则性与容量性质。

$C-$Robin Functions and Applications

We continue the study in the setting of pluripotential theory arising from polynomials associated to a convex body $C$ in $({\bf R}^+)^d$. Here we discuss $C-$Robin functions and their applications. In the particular case where $C$ is a simplex in $({\bf R}^+)^2$ with vertices $(0,0),(b,0),(a,0)$, $a,b>0$, we generalize results of T. Bloom to construct families of polynomials which recover the $C-$extremal function $V_{C,K}$ of a nonpluripolar compact set $K\subset {\bf C}^d$.

The Extremal Function for the Complex Ball for Generalized Notions of Degree and Multivariate Polynomial Approximation

We discuss the Siciak-Zaharjuta extremal function of pluripotential theory for the unit ball in C^d for spaces of polynomials with the notion of degree determined by a convex body P. We then use it to analyze the approximation properties of such polynomial spaces, and how these may differ depending on the function f to be approximated.

A Hilbert Lemniscate Theorem in C^2

For a regular, compact, polynomially convex circled set K in C^2, we construct a sequence of pairs {P_n,Q_n} of homogeneous polynomials in two variables with deg P_n = deg Q_n = n such that the sets K_n: = {(z,w) \in C^2 : |P_n(z,w)| \leq 1, |Q_n(z,w)| \leq 1} approximate K and the normalized counting measures {μ_n} associated to the finite set {P_n = Q_n = 1} converge to the pluripotential-theoretic Monge-Ampere measure for K. The key ingredient is an approximation theorem for subharmonic functions of logarithmic growth in one complex variable.

On the Convergence of Optimal Measures

Using recent results of Berman and Boucksom we show that for a non-pluripolar compact set K in C^d and an admissible weight function w=e^{-ϕ} any sequence of so-called optimal measures converges weak-* to the equilibrium measure μ_{K,ϕ} of (weighted) Pluripotential Theory for K,ϕ.

椭圆与加权Szego投影的多项式到多项式映射

Ellipses and Polynomial-to-Polynomial Mapping of Weighted Szego Projections

该研究结合位势理论相关结果，探究椭圆与椭球上的加权Szego投影，证明平面椭圆上存在能将多项式映射为同次数多项式的加权Szego投影。

圆弧上的加权残差多项式

Weighted residual polynomials on a circular arc

本文研究了圆弧上加权残差多项式的行为，推导了Widom因子的渐进行为并应用于lemniscatic圆弧。

Bernstein's problem on weighted polynomial approximation

We formulate and discuss a necessary and sufficient condition for polynomials to be dense in a space of continuous functions on the real line, with respect to Bernstein's weighted uniform norm. Equivalently, for a positive finite measure $μ$ on the real line we give a criterion for density of polynomials in $L^p(μ)$.

复平面紧子集上的加权切比雪夫多项式

Weighted Chebyshev Polynomials on Compact Subsets of the Complex Plane

本文研究复平面紧子集上关于有界权函数的加权切比雪夫多项式，建立了其存在唯一性，导出了多个经典准则的加权类比，并证明了 Widom 因子的不变性及基于 Szegő 积分的下界。

Weighted-$L^2$ polynomial approximation in $\mathbb{C}$

We study the density of polynomials in $H^2(Ω,e^{-φ})$, the space of square integrable holomorphic functions in a bounded domain $Ω$ in $\mathbb{C}$, where $φ$ is a subharmonic function. In particular, we prove that the density holds in Carathéodory domains for any subharmonic function $φ$ in a neighborhood of $\overlineΩ$. In non-Carathéodory domains, we prove that the density depends on the weight function, giving examples.

具有指数在紧凸集中的多项式及相关的加权极值函数——广义乘积性质

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Generalized product property

本文推广了Siciak关于两个集合的极值函数关系的结果，探讨了在紧凸集增长条件下Siciak-Zakharyuta函数的广义乘积性质及其在加权情况下的局限性。

指数位于紧凸集中的多项式及相关加权极值函数——Bernstein-Walsh-Siciak定理

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- The Bernstein-Walsh-Siciak theorem

本文将Bernstein-Walsh-Siciak定理推广至指数受限于紧凸集$mS$的多项式子环，并以容许权函数下的加权一致估计替代普通一致误差估计。

具有在紧凸集中的指数的多项式及相关的加权极值函数 -- 近似与正则性

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Approximations and regularity

本文研究了与紧凸集相关的多项式及加权极值函数的正则性问题，通过积分、极小和极大连算子证明了相关函数的下半连续性，并推翻了Hölder连续性的已有结论。

通过加权多项式逼近得到泛函泰勒级数的新结果

New results on universal Taylor series via weighted polynomial approximation

本文利用加权多项式逼近证明了具有非空内部的紧集上泛函泰勒级数的发散性与稠密性，改进了Mouze的结果，构造了非零泰勒系数模趋于无穷的全纯函数，并补充了Pritsker和Varga关于加权多项式逼近的条件。

Polynomial approximation in weighted Dirichlet spaces

We give an elementary proof of an analogue of Fejér's theorem in weighted Dirichlet spaces with superharmonic weights. This provides a simple way of seeing that polynomials are dense in such spaces.

Topology of polar weighted homogeneous hypersurfaces

Polar weighted homogeneous polynomials are the class of special polynomials of real variables $x_i,y_i, i=1,..., n$ with $z_i=x_i+\sqrt{-1} y_i$, which enjoys a "polar action". In many aspects, their behavior looks like that of complex weighted homogeneous polynomials. We study basic properties of hypersurfaces which are defined by polar weighted homogeneous polynomials.

Approximation by polynomials in a weighted space of infinitely differentiable functions

The density of polynomials in a weighted space of infinitely differentiable functions in a multidimensional real space is proved under minimal conditions on weight functions and on differences between weight functions. We apply this result for description of strong dual for weighted spaces of infinitely differentiable functions on real line and weighted spaces of sequences of infinitely differentiable functions on real line in terms of the Fourier-Laplace transform of functionals.

Bergman kernels for weighted polynomials and weighted equilibrium measures of C^n

Various convergence results for the Bergman kernel of the Hilbert space of all polynomials in \C^{n} of total degree at most k, equipped with a weighted norm, are obtained. The weight function is assumed to be C^{1,1}, i.e. it is differentiable and all of its first partial derivatives are locally Lipshitz continuous. The convergence is studied in the large k limit and is expressed in terms of the global equilibrium potential associated to the weight function, as well as in terms of the Monge-Ampere measure of the weight function itself on a certain set. A setting of polynomials associated to a…

Weighted $L^2$-Norms of Gegenbauer polynomials

We study integrals of the form \begin{equation*} \int_{-1}^1(C_n^{(λ)}(x))^2(1-x)^α(1+x)^β\, dx, \end{equation*} where $C_n^{(λ)}$ denotes the Gegenbauer-polynomial of index $λ>0$ and $α,β>-1$. We give exact formulas for the integrals and their generating functions, and obtain asymptotic formulas as $n\to\infty$.

Weighted holomorphic Dirichlet series and composition operators with polynomial symbols

In this paper, we introduce a general class of weighted spaces of holomorphic Dirichlet series (with real frequencies) analytic in some half-plane and study composition operators on these spaces. In the particular case when the symbol inducing the composition operator is an affine function, we give criteria for boundedness and compactness. We also study the cyclicity property and as a byproduct give a sufficient condition so that the direct sum of the identity plus a weighted forward shift operator on the Hardy space H^2 is cyclic.

Uniform Convolution and Fourier Restriction estimates for complex polynomial curves in $\mathbb{C}^3$

We establish optimal $(p,q)$ ranges for two types of estimates associated to three dimensional complex polynomial curves. These are the estimates for the weighted restriction of the Fourier Transform to a complex polynomial curve, and the weighted Convolution Operator associated to a complex polynomial curve. Establishing these estimates comes down to establishing a lower bound for the Jacobian of a mapping associated to the complex curve in question.

Polynomial approach to cyclicity for weighted $\ell^p_A$

In previous works, an approach to the study of cyclic functions in reproducing kernel Hilbert spaces has been presented, based on the study of so called \emph{optimal polynomial approximants}. In the present article, we extend such approach to the (non-Hilbert) case of spaces of analytic functions whose Taylor coefficients are in $\ell^p(ω)$, for some weight $ω$. When $ω=\{(k+1)^α\}_{k\in \mathbb{N}}$, for a fixed $α\in \mathbb{R}$, we derive a characterization of the cyclicity of polynomial functions and, when $1<p<\infty$, we obtain sharp rates of convergence of the optimal norms.