Zero Distributions for Polynomials Orthogonal with Weights over Certain Planar Regions
Zero Distributions for Polynomials Orthogonal with Weights over Certain Planar Regions
复制标题
某些平面区域上与权重正交的多项式的零分布
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
N. Stylianopoulos
中科院分区：
文献类型：
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作者：
E. Miña;E. Saff;N. Stylianopoulos
设G是G中的有界Jordan域，wn = 0是G上的解析函数，使得tSGs ε ω ε 2dm < ∞，其中dm是面积测度.本文研究了G上关于ω ω 2dm正交的多项式列的零点分布。我们发现这样的分布依赖于G上解析空间Lω2（G）：= f的再生核Kw（z，ω）的奇点位置：<$G <$f <$2 <$ω <$2dm < ∞.对于Kω（·，ω）在G上有奇点的情形，给出了一个基本定理.为了研究相反的情况，我们详细考虑两个例子：第一个例子是当G是单位圆盘并且ω是亚纯的，第二个例子是当G是透镜形区域并且ω是整的。我们的分析也可以应用于当G是一个矩形或特殊三角形的情况下的ω 1。我们还提供了Kω（·，ω）的公式，这些公式有助于确定它的奇性。
Let G be a bounded Jordan domain in ℂ and let w n = 0 be an analytic function on G such that tSGs¦ω¦2dm < ∞, where dm is the area measure. We investigate the zero distribution of the sequence of polynomials that are orthogonal on G with respect to ¦ω¦2dm. We find that such a distribution depends on the location of the singularities of the reproducing kernel Kw(z, ζ) of the space Lω2(G):= f analytic on G: ∫G¦ f ¦2¦ω¦2dm < ∞. A fundamental theorem is given for the case when Kω(·, ζ) has a singularity on ∂G for at least some ζ ∈ G. To investigate the opposite case, we consider two examples in detail: first when G is the unit disk and ω is meromorphic, and second when G is a lens-shaped domain and ω is entire. Our analysis can also be applied for ω ≡ 1 in the case when G is a rectangle or a special triangle. We also provide formulas for Kω(·, ζ) that are of help for the determination of its singularities.