齐性集上正交多项式的渐近行为
Asymptotic behavior of polynomials orthonormal on a homogeneous set
AI总结：
本文研究齐性紧集上正交多项式的渐近行为，在 Szegö 型条件下给出支集内外的渐近表示，并证明递推系数渐近概周期，推广了 Widom、Szegö 及 Kolmogorov--Krein 的经典结果。
摘要：
设 $E$ 为齐性紧集，例如具有正长度的康托集。进一步设 $σ$ 为满足 $\ ext{supp}(σ)=E$ 的正测度。在 $σ$ 的绝对连续部分满足 Szegö 型条件的假设下，我们给出了关于 $σ$ 的正交多项式在支集上及支集外的渐近表示。对于 $E$ 由有限个区间组成且 $σ$ 无奇异分量的特殊情况，这是 Widom 的一个众所周知的结果。若 $E=[a,b]$，则成为 Szegö 的经典结果；若还出现奇异分量，则归功于 Kolmogorov--Krein。事实上，结果是在更一般的情形下给出的：正交性测度在 $E$ 外可具有可数个质量点，这些质量点假定仅聚于 $E$，并且（连同相关的 Stieltjes 函数的零点）满足自由插值 Carleson 型条件。即使对于单区间情形，除有限个质量点的情况外，这一结果也是新的。此外，作为我们表示的副产品，我们得到正交多项式的递推系数具有渐近概周期性。换言之，与上述正交多项式相关的 Jacobi 矩阵是谱为齐性的概周期 Jacobi 矩阵的单侧限制的紧扰动。我们的主要工具是 Widom 型黎曼曲面上特征自守函数与形式的 Hardy 空间理论，同时也使用一维 Schrödinger 方程散射理论的一些思想。
英文摘要：
Let $E$ be a homogeneous compact set, for instance a Cantor set of positive length. Further let $σ$ be a positive measure with $\text{supp}(σ)=E$. Under the condition that the absolutely continuous part of $σ$ satisfies a Szegö--type condition we give an asymptotic representation, on and off the support, for the polynomials orthonormal with respect to $σ$. For the special case that $E$ consists of a finite number of intervals and that $σ$ has no singular component this is a nowaday well known result of Widom. If $E=[a,b]$ it becomes a classical result due to Szegö and in case that there appears in addition a singular component, it is due to Kolmogorov--Krein. In fact the results are presented for the more general case that the orthogonality measure may have a denumerable set of mass--points outside of $E$ which are supposed to accumulate on $E$ only and to satisfy (together with the zeros of the associated Stieltjes function) the free--interpolation Carleson--type condition. Up to the case of a finite number of mass points this is even new for the single interval case. Furthermore, as a byproduct of our representations, we obtain that the recurrence coefficients of the orthonormal polynomials behave asymptotically almost periodic. Or in other words the Jacobi matrices associated with the above discussed orthonormal polynomials are compact perturbations of a one--sided restriction of almost periodic Jacobi matrices with homogeneous spectrum. Our main tool is a theory of Hardy spaces of character--automorphic functions and forms on Riemann surfaces of Widom type, we use also some ideas of scattering theory for one--dimensional Schrödinger equations.
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