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

Abstract

In this paper we investigate greedy energy sequences on the unit circle for the logarithmic and Riesz potentials. By definition, if (an) ∞ n=0 is a greedy s-energy sequence on the unit circle, the Riesz potential UN,s(x) := N−1 k=0 |a k − x| −s , s > 0, generated by the first N points of the sequence attains its minimum value on the circle at the point aN . In this work, we analyze the asymptotic properties of these extremal values UN,s(aN ), treating separately the cases 0 < s < 1, s = 1, and s > 1. We present new second-order asymptotic formulas for UN,s(aN ) in the cases 0 < s < 1 and s = 1. A first-order result for s > 1 is proved, and it is shown that the first-order normalized sequence (UN,s(aN )/N s ) ∞ N=1 is divergent in this case.