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

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.

Asymptotics of the minimum values of Riesz and logarithmic potentials generated by greedy energy sequences on the unit circle

“…In this work we focus on the particular case of the unit circle K = S 1 , and analyze in detail the asymptotic behavior of the extremal values U N,s (a N ) in (1.3), for all values of s > 0. Our results complement those obtained in [20], which described in this context the asymptotic behavior of the energies E s (α N,s ). We also refine the asymptotic formula (1.2) for the polynomials P n .…”

“…In [2], Proposition 3.4 was proved for Leja sequences. But as it was remarked in [20], this result holds for greedy s-energy sequences as well, for any value of s > 0, since greedy s-energy sequences coincide geometrically with Leja sequences on S 1 . We now state this result in terms of greedy s-energy sequences on S 1 .…”

“…It was proved in [14] that for any s > 0, every N -point minimal s-energy configuration ω N,s on S 1 consists of N equally spaced points, and in [5] a complete asymptotic expansion in N of E s (ω N,s ) was found. The analyses in [19,20] revealed how the asymptotic behaviors of the quantities E s (α N,s ) and E s (ω N,s ) differ significantly when we look at first-order asymptotics in the range s > dim S 1 = 1, and they also differ for second-order asymptotics in the range 0 < s ≤ 1, in contrast to the similarities described above.…”

Polynomials With Zeros on the Unit Circle: Regularity of Leja Sequences

“…Pausinger [35] characterized the arising sequences (since there are often several maxima, there is an ambiguity in which one to pick and one can obtain several sequences); he also showed that this characterization also holds true for a larger class of notions of energy. Lopez‐Garcia and Wagner [34] established energy asymptotics. Götz [25], building on earlier machinery of Andrievski and Blatt [3, 4], Blatt [14], Blatt and Mhaskar [15], and Totik [45], proved that .…”

Green energy and extremal decompositions

“…Introduction and statement of results. There are many studies related to the extremal problems for different kinds of energy of a discrete charge (see, e. g., the papers [2][3][4], [9], [10], [12], and the references therein). In contrast to the previous research, we consider the Green energy (see also the recent articles [1] and [5]).…”