AI 中文总结
本文研究Riesz框架下的约束平衡问题，给出一般性结果，详细分析单位球中该问题的示例解，并通过数值实验说明如何用约束Leja点离散化约束平衡测度。
AI 中文摘要
Rakhmanov于1996年提出了对数势的约束平衡问题，因为他意识到紧区间上离散正交多项式零点的渐近分布可通过该区间在满足特定约束的测度类中的平衡测度来描述。Dragnev和Saff（1997）、Kuijlaars和Van Assche（1999）等学者将该方法推广到更一般的情形，这些问题已被证明可用于描述不同场景下的渐近分布。本文研究Riesz框架下的约束平衡问题，即针对d维欧氏空间ℝᵈ（d≥1）中超平面上的s-Riesz势，其中max(0,d-2)<s<d。除了一些一般性结果外，本文还详细研究了一个由单位球中约束平衡问题的解构成的示例，这是本文的核心部分。最后，大量数值实验表明，可利用Coroian和Dragnev于2001年提出的所谓“约束Leja点”，将问题的解——约束平衡测度进行离散化处理。
英文摘要
The constrained equilibrium problem for the logarithmic potential was introduced by Rakhmanov (1996) as he realized that the asymptotic distribution of the zeros of discrete orthogonal polynomials in a compact interval could be described in terms of the equilibrium measure of this interval in a class of measures subject to a certain constraint. Other authors such as Dragnev and Saff (1997), and Kuijlaars and Van Assche (1999), extended this approach to more general settings. These problems have proven to be useful for describing asymptotic distributions in different settings. In the current paper, we consider constrained equilibrium problems in the Riesz setting, that is, for $s$-Riesz potentials in the hyperplane $\mathbb{R}^d,$ with $d\geq 1$ and $\max (0,d-2) < s < d$. Along with some general results, an illustrative example consisting of the solution of a constrained equilibrium problem in the unit ball is studied in detail. This is the main part of the paper. Finally, a number of numerical experiments show how the constrained equilibrium measure, the solution of the problem, may be discretized using the so-called 'constrained Leja points' introduced by Coroian and Dragnev (2001).
Comments19 pages, 10 figures
