Orthogonal Polynomials Defined by Self-Similar Measures with Overlaps

New Preprint by Sze-man Ngai:

Orthogonal Polynomials Defined by Self-Similar Measures with Overlaps

SZE-MAN NGAI, WEI TANG, ANH TRAN, AND SHUAI YUAN

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 coeffi- cients 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.

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