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

Journal Article · · Experimental mathematics
 [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)
+ Show Author Affiliations

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.

Research Organization:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA); National Natural Science Foundation of China (NNSCF)
Grant/Contract Number:
AC04-94AL85000; 11771136; 1127122; 11901187; NA0003525
OSTI ID:
1614785
Report Number(s):
SAND-2020-3554J; 685001
Journal Information:
Experimental mathematics, Vol. 31, Issue 3; ISSN 1058-6458
Publisher:
Taylor & FrancisCopyright Statement
Country of Publication:
United States
Language:
English

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Related Subjects

97 MATHEMATICS AND COMPUTING
orthogonal polynomial
self-similar measure with overlaps
Nevai class