Orthogonal polynomials defined by self-similar measures with overlaps

Sze-Man Ngai Georgia Southern University and Hunan Normal University Wei Tang Hunan First Normal University Anh Tran Georgia Institute of Technology Shuai Yuan University of South Carolina Columbia

Publications of CMSA of Harvard mathscidoc:1907.38001

We study orthogonal polynomials with respect to self-similar measures, focusing on the class of in finite 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.
Orthogonal polynomial, self-similar measure with overlaps, Nevai class
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@inproceedings{sze-manorthogonal,
  title={Orthogonal polynomials defined by self-similar measures with overlaps},
  author={Sze-Man Ngai, Wei Tang, Anh Tran, and Shuai Yuan},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190703224511468879397},
}
Sze-Man Ngai, Wei Tang, Anh Tran, and Shuai Yuan. Orthogonal polynomials defined by self-similar measures with overlaps. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190703224511468879397.
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