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.
2010 Mathematics Subject Classification:
Acknowledgments
Part of this work was carried out while the first author was visiting the Center of Mathematical Sciences and Applications of Harvard University. He is indebted to Professor Shing-Tung Yau for the opportunity to visit the center and thanks the center for its hospitality and support. Another part of this work was performed while AT and SY were graduate students in Georgia Southern University. The authors thank Professor Mantica for some helpful conversations. They are also grateful for some valuable comments and suggestions from the anonymous reviewers. The views expressed in the article do not necessarily represent the views of the U.S. Department of Energy or the United States Government. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.