Asymptotics of the Energy of Sections of Greedy Energy Sequences on the Unit Circle, and Some Conjectures for General Sequences Articles uri icon

authors

  • LOPEZ GARCIA, ABEY
  • Wagner, Douglas

publication date

  • December 2015

start page

  • 721

end page

  • 750

issue

  • 4

volume

  • 15

International Standard Serial Number (ISSN)

  • 1617-9447

Electronic International Standard Serial Number (EISSN)

  • 2195-3724

abstract

  • Abstract In this paper we investigate the asymptotic behavior of the Riesz s-energy of the first N points of a greedy s-energy sequence on the unit circle, for all values of s in the range 0 ≤ s < ∞ (identifying as usual the case s = 0 with the logarithmic energy). In the context of the unit circle, greedy s-energy sequences coincide with the classical Leja sequences constructed using the logarithmic potential. We obtain first-order and second-order asymptotic results. The key idea is to express the Riesz s-energy of the first N points of a greedy s-energy sequence in terms of the binary representation of N. Motivated by our results, we pose some conjectures for general sequences on the unit circle.

keywords

  • greedy energy sequence; leja sequence; riesz kernel; equilibrium; measure; optimal energy configuration