Abstract

We study orthogonal polynomials with respect to self-similar measures, focusing on the class of infinite Bernoulli convolutions, which are defined by iterated function systems with overlaps, especially those defined by the Pisot, Garsia, and Salem numbers. By using an algorithm of Mantica, we obtain graphs of the coefficients of the 3-term recursion relation defining the orthogonal polynomials. We use these graphs to predict whether the singular infinite Bernoulli convolutions belong to the Nevai class. Based on our numerical results, we conjecture that all infinite Bernoulli convolutions with contraction ratios greater than or equal to 1/2 belong to Nevai’s class, regardless of the probability weights assigned to the self-similar measures.

摘要

我们研究关于自相似度量的正交多项式，重点是无限伯努利卷积类，它由具有重叠的迭代函数系统定义，特别是由 Pisot、Garsia 和 Salem 数定义的函数系统。通过使用 Mantica 的算法，我们获得了定义正交多项式的 3 项递归关系的系数图。我们使用这些图来预测奇异无限伯努利卷积是否属于 Nevai 类。根据我们的数值结果，我们推测所有收缩率大于或等于 1/2 的无限伯努利卷积都属于 Nevai 类，无论分配给自相似度量的概率权重如何。