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arXiv · 0912.4720

Discrete Energy Asymptotics on a Riemannian circle

J. S. Brauchart·D. P. Hardin·E. B. Saff
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Abstract

We derive the complete asymptotic expansion in terms of powers of $N$ for the geodesic $f$-energy of $N$ equally spaced points on a rectifiable simple closed curve $Γ$ in ${\mathbb R}^p$, $p\geq2$, as $N \to \infty$. For $f$ decreasing and convex, such a point configuration minimizes the $f$-energy $\sum_{j\neq k}f(d(\mathbf{x}_j, \mathbf{x}_k))$, where $d$ is the geodesic distance (with respect to $Γ$) between points on $Γ$. Completely monotonic functions, analytic kernel functions, Laurent series, and weighted kernel functions $f$ are studied. % Of particular interest are the geodesic Riesz potential $1/d^s$ ($s \neq 0$) and the geodesic logarithmic potential $\log(1/d)$. By analytic continuation we deduce the expansion for all complex values of $s$.

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J. S. Brauchart, D. P. Hardin, E. B. Saff. 2009-12-23. Discrete Energy Asymptotics on a Riemannian circle. https://arxiv.org/abs/0912.4720

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