Discrete Energy Asymptotics on a Riemannian circle

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 $\Gamma$ in ${\mathbb{R}}^p$, $p\ge 2$, as $N\to \infty$. For $f$ decreasing and convex, such a point configuration minimizes the $f$-energy ${\sum }_{j\ne k}f(d({\mathbf{x}}_j,{\mathbf{x}}_k))$, where $d$ is the geodesic distance (with respect to $\Gamma$) between points on $\Gamma$. Completely monotonic functions, analytic kernel functions, Laurent series, and weighted kernel functions $f$ are studied. % Of particular interest are the geodesic Riesz potential $1/d^s$ ($s\ne 0$) 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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