论文标题
正交多项式的伯格曼内核的极端问题
An extremal problem for the Bergman kernel of orthogonal polynomials
论文作者
论文摘要
令$γ\ subset \ mathbb c $为$ c（2，α）$的曲线。对于$ Z_ {0} $，在$ {\ Mathbb C} \setMinusγ$的无限制组件中，对于$ n = 1,2，... $，让$ν_n$是supp $（ν_{n}）\ subsetγ$的概率度量$ b_ {n}（ν，z）：= \ sum_ {k = 0}^{n} | q_ {k}^ν（z）|^{2} $ at $ z_ {0} $在$ z_ {0} $中$ \ {q_ {0}^ν，\ ldots，q_ {n}^ν\} $是$ l^2（ν）$在最多$ n $的$ l^2（ν）$中的正常基础）。我们表明，$ \ {ν_{n} \} _ n $趋于弱 - *至$ \hatδ_{z_ {0}} $，$ z_0 $ point质量的balayage to $ z_0 $ to $γ$，通过将其与单位圆的优化问题相关联。我们的证明利用了与$γ$相关的Faber多项式的估计值。
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 $Γ$.
