Orthogonal Polynomials Defined by Self-Similar Measures with Overlaps

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.

Keywords:

• Orthogonal polynomial
• self-similar measure with overlaps
• Nevai class

2010 Mathematics Subject Classification:

• Primary: 28A80
• 42C05
• Secondary: 33C45
• 28A25

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.

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Figure 1. Figure showing the IFS defining the 3-fold convolution.

Additional information

Funding

The first two authors are supported in part by the National Natural Science Foundation of China, Grant nos. 11771136 and 11271122, and Construct Program of the Key Discipline in Hunan Province. The first author is also supported in part by the Hunan Province Hundred Talents Program, the Center of Mathematical Sciences and Applications (CMSA) of Harvard University. The second author is also supported in part by the National Natural Science Foundation of China, Grant no. 11901187. SN, AT, and SY are also supported in part by a Faculty Research Scholarly Pursuit Award from Georgia Southern University.