单位圆上正交多项式及其导数

ORTHOGONAL POL YNOMIALS ON THE UNIT CIRCLE AND THEIR DERIVATIVES

对单位圆上关于有限正Ｂｏｒｅｌ测度的正交多项式导数的渐近性质的研究在七十年代已有所结果和突破，如熟知的Ｓｚｅｇ理论等。然而对其微分性质的分析和讨论并不算多，即使如此，也只是限于考虑单位圆上正交多项式的某些特殊类型￣［１］。本文证明单位圆上正交多项式序列的导数仍然是单位圆上正交多项式序列，并给出它们与相关微分方程之间的一些关系。

We characterize the classical families in the Hahn sense.The seqquence of derivatives of orthogonal polynomials on the unit circle constitute a sequence of orthogonal polynomials on the unit circle iff the Toeplitz matrix for the moments is diagonal smoe relations for the sequences of derivatives of orthogonal preserving maps and differential equations for orthogonal polynomials on the unit circle.