首页/论文/2003.12833

arXiv 2003.128332020-03math.CA

Geometric computation of Christoffel functions on planar convex domains

A. Prymak

发表信息：: Journal of Approximation Theory 268 (2021) 105603
DOI：: 10.1016/j.jat.2021.105603

英文摘要：

For arbitrary planar convex domain, we compute the behavior of Christoffel function up to a constant factor using comparison with other simple reference domains. The lower bound is obtained by constructing an appropriate ellipse contained in the domain, while for the upper bound an appropriate parallelepiped containing the domain is constructed. As an application we obtain a new proof of existence of optimal polynomial meshes in planar convex domains.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Pointwise behavior of Christoffel function on planar convex domains

We prove a general lower bound on Christoffel function on planar convex domains in terms of a modification of the parallel section function of the domain. For a certain class of planar convex domains, in combination with a recent general upper bound, this allows to compute the pointwise behavior of Christoffel function. We illustrate this approach for the domains $\{(x,y):|x|^α+|y|^α\le1\}$, $1<α<2$, and compute up to a constant factor the required modification of the parallel section function, and, consequently, Christoffel function at an arbitrary interior point of the domain.

2017-09-29math.CA

Christoffel function on planar domains with piecewise smooth boundary

We compute up to a constant factor the Christoffel function on planar domains with boundary consisting of finitely many $C^2$ curves such that each corner point of the boundary has interior angle strictly between $0$ and $π$. The resulting formula uses the distances from the point of interest to the curves or certain parts of the curves defining the boundary of the domain.

2019-02-18math.CA

Upper estimates of Christoffel function on convex domains

New upper bounds on the pointwise behaviour of Christoffel function on convex domains in ${\mathbb{R}}^d$ are obtained. These estimates are established by explicitly constructing the corresponding "needle"-like algebraic polynomials having small integral norm on the domain, and are stated in terms of few easy-to-measure geometric characteristics of the location of the point of interest in the domain. Sharpness of the results is shown and examples of applications are given.

2017-06-30math.CA

Strong asymptotics for Christoffel functions of planar measures

We prove a version of strong asymptotics of Christoffel functions with varying weights for a general class of sets E and measures in the complex plane. This class includes all regular measures in the sense of Stahl-Totik on regular compact sets E in the plane and even allows varying weights. Our main theorems cover some known results for subsets E of the real line R; in particular, we recover information in the case of E=R with Lebesgue measure dx and weight w(x) = exp(-Q(x)) where Q(x) is a nonnegative, even degree polynomial having positive leading coefficient.

2007-09-13math.CAmath.CV

Fuglede conjecture holds for convex planar domains

Let $Ω$ be a compact convex domain in the plane. We prove that $L^2(Ω)$ has an orthogonal basis of exponentials if and only if $Ω$ tiles the plane by translation.

2001-04-07math.CA

Rank-one convexity implies polyconvexity in isotropic planar incompressible elasticity

We study convexity properties of energy functions in plane nonlinear elasticity of incompressible materials and show that rank-one convexity of an objective and isotropic elastic energy $W$ on the special linear group $\mathrm{SL}(2)$ implies the polyconvexity of $W$.

2016-09-05math.CA

更多相关论文收起更多论文

严格凸平面图上的一致完美测度是$L^2$-平坦的

Uniformly perfect measures on strictly convex planar graphs are $L^{2}$-flattening

本文针对严格凸平面$C^2$图上的一致完美测度展开研究，证明该类测度均满足$L^2$平坦性质，给出了对应傅里叶变换的$L^p$范数衰减估计。

2025-11-16math.CA

Distances sets that are a shift of the integers and Fourier basis for planar convex sets

The aim of this paper is to prove that if a planar set $A$ has a difference set $Δ(A)$ satisfying $Δ(A)\subset \Z^++s$ for suitable $s$ than $A$ has at most 3 elements. This result is motivated by the conjecture that the disk has not more than 3 orthogonal exponentials. Further, we prove that if $A$ is a set of exponentials mutually orthogonal with respect to any symmetric convex set $K$ in the plane with a smooth boundary and everywhere non-vanishing curvature, then $ # (A \cap {[-q,q]}^2) \leq C(K) q$ where $C(K)$ is a constant depending only on $K$. This extends and clarifies in the plane t…

2007-09-26math.CA

New $L^p$ bounds for Bochner-Riesz multipliers associated with convex planar domains with rough boundary

We consider generalized Bochner-Riesz multipliers of the form $(1-ρ(ξ))_+^λ$ where $ρ(ξ)$ is the Minkowski functional of a convex domain in $\mathbb{R}^2$, with emphasis on domains for which the usual Carleson-Sjölin $L^p$ bounds can be improved. We produce convex domains for which previous results due to Seeger and Ziesler are not sharp. We identify two key properties of convex domains that lead to improved $L^p$ bounds for the associated Bochner-Riesz operators. First, we introduce the notion of the "additive energy" of the boundary of a convex domain. Second, we associate a set of direction…

2016-10-11math.CA

Lattice points in large convex planar domains of finite type

Let $\mathcal{B}$ be a compact convex planar domain with smooth boundary of finite type and $\mathcal{B}_θ$ its rotation by an angle $θ$. We prove that for almost every $θ\in[0, 2π]$ the remainder $P_{\mathcal{B}_θ}(t)$ is of order $O_θ(t^{2/3-ζ})$ with a positive number $ζ$ independent of the domain.

2011-06-01math.CAmath.NT

The Euler Equations in planar nonsmooth convex domains

We consider the Euler system set on a bounded convex planar domain, endowed with impermeability boundary conditions. This system is a model for the barotropic mode of the Primitive Equations on a rectangular domain. We show the existence of weak solutions with L^p vorticity for 4/3<= p <= 2, extending and enriching a previous result of Taylor. In the physically interesting case of a rectangular domain, a similar result holds for all 2<p<\infty as well. Moreover, we show the uniqueness of solutions with bounded initial vorticity. The main tool is a new BMO regularity estimate for the Dirichlet …

2013-08-16math.APmath.CA

Confirmation of Matheron's conjecture on the covariogram of a planar convex body

The covariogram g_K of a convex body K in E^d is the function which associates to each x in E^d the volume of the intersection of K with K+x. In 1986 G. Matheron conjectured that for d=2 the covariogram g_K determines K within the class of all planar convex bodies, up to translations and reflections in a point. This problem is equivalent to some problems in stochastic geometry and probability as well as to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffrac…

2007-11-30math.CAmath.MGmath.PR

平面乌拉姆浮体问题中周长密度σ=1/6的刚性

Rigidity in the Planar Ulam Floating Body Problem with perimetral density $σ=\tfrac16$

University of Cincinnati（辛辛那提大学）

研究平面凸域在周长密度σ=1/6下的乌拉姆浮体问题，利用Zindler旋转装置框架，将问题转化为与内接正六边形相关的二维动力系统。主要结果表明，圆盘是唯一在该周长密度下在任意位置都能保持平衡的凸域，为有理周长密度的凸设定提供了新的刚性结果。

2026-04-11math.CAmath.MG