TY - JOUR

T1 - Orthogonal Polynomials Defined by Self-Similar Measures with Overlaps

AU - Ngai, Sze Man

AU - Tang, Wei

AU - Tran, Anh

AU - Yuan, Shuai

N1 - Publisher Copyright:
© 2020 Taylor & Francis Group, LLC.

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

KW - Nevai class

KW - Orthogonal polynomial

KW - self-similar measure with overlaps

UR - http://www.scopus.com/inward/record.url?scp=85083555190&partnerID=8YFLogxK

U2 - 10.1080/10586458.2020.1743214

DO - 10.1080/10586458.2020.1743214

M3 - Article

AN - SCOPUS:85083555190

SN - 1058-6458

VL - 31

SP - 1026

EP - 1038

JO - Experimental Mathematics

JF - Experimental Mathematics

IS - 3

ER -