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Title: Orthogonal Polynomials Defined by Self-Similar Measures with Overlaps

Abstract

Here, 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.

Authors:
 [1];  [2];  [3];  [4]
  1. Hunan Normal Univ., Changsha (China); Georgia Southern Univ., Statesboro, GA (United States)
  2. Hunan First Normal Univ., Changsha (China)
  3. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  4. Univ. of South Carolina, Columbia, SC (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA); National Natural Science Foundation of China (NNSCF)
OSTI Identifier:
1614785
Report Number(s):
SAND-2020-3554J
Journal ID: ISSN 1058-6458; 685001
Grant/Contract Number:  
AC04-94AL85000; 11771136; 1127122; 11901187; NA0003525
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Experimental mathematics
Additional Journal Information:
Journal Volume: 31; Journal Issue: 3; Journal ID: ISSN 1058-6458
Publisher:
Taylor & Francis
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; orthogonal polynomial; self-similar measure with overlaps; Nevai class

Citation Formats

Ngai, Sze-Man, Tang, Wei, Tran, Anh, and Yuan, Shuai. Orthogonal Polynomials Defined by Self-Similar Measures with Overlaps. United States: N. p., 2020. Web. doi:10.1080/10586458.2020.1743214.
Ngai, Sze-Man, Tang, Wei, Tran, Anh, & Yuan, Shuai. Orthogonal Polynomials Defined by Self-Similar Measures with Overlaps. United States. https://doi.org/10.1080/10586458.2020.1743214
Ngai, Sze-Man, Tang, Wei, Tran, Anh, and Yuan, Shuai. 2020. "Orthogonal Polynomials Defined by Self-Similar Measures with Overlaps". United States. https://doi.org/10.1080/10586458.2020.1743214. https://www.osti.gov/servlets/purl/1614785.
@article{osti_1614785,
title = {Orthogonal Polynomials Defined by Self-Similar Measures with Overlaps},
author = {Ngai, Sze-Man and Tang, Wei and Tran, Anh and Yuan, Shuai},
abstractNote = {Here, 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.},
doi = {10.1080/10586458.2020.1743214},
url = {https://www.osti.gov/biblio/1614785}, journal = {Experimental mathematics},
issn = {1058-6458},
number = 3,
volume = 31,
place = {United States},
year = {Fri Apr 10 00:00:00 EDT 2020},
month = {Fri Apr 10 00:00:00 EDT 2020}
}

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