Asymptotics of greedy energy sequences on the unit circle and the sphere

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:504

Article

Lopez-Garcia, Abey¹  McCleary, Ryan E.¹ 

[1] Univ Cent Florida, Dept Math, 4393 Andromeda Loop North, Orlando, FL 32816 USA

关键词: Greedy lambda-energy sequence;    Leja sequence;    Maximal distribution;    Potential;    Binary representation;    Riemann zeta function;   

DOI  :  10.1016/j.jmaa.2021.125269

来源: Elsevier

【 摘 要 】

For a parameter lambda > 0, we investigate greedy lambda-energy sequences (a(n))(n=0)(infinity) on the unit sphere S-d subset of Rd+1, d >= 1, satisfying the defining property that each a(n), n >= 1, is a point where the potential Sigma(n-1)(k=0)vertical bar x - a(k)vertical bar(lambda) attains its maximum value on S-d. We show that these sequences satisfy the symmetry property a(2k)(+1) = -a(2k) for every k >= 0. The asymptotic distribution of the sequence undergoes a sharp transition at the value lambda = 2, from uniform distribution (lambda < 2) to concentration on two antipodal points (lambda > 2). We investigate first-order and second order asymptotics of the lambda-energy of the first N points of the sequence, as well as the asymptotic behavior of the extremal values Sigma(n-1)(k=0)vertical bar a(n) - a(k)vertical bar(lambda). The second-order asymptotics is analyzed on the unit circle. It is shown that this asymptotic behavior differs significantly from that of N equally spaced points on the unit circle, and a transition in the behavior takes place at lambda = 1. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

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