Discrete Energy Asymptotics on a Riemannian circle

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 $Rset^p$, $p≥2$, as $N o infty$. For $f$ decreasing and convex, such a point configuration minimizes the $f$-energy $sum_{j eq k}f(d(mathbf{x}_j, mathbf{x}_k))$, 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 $1/d^s$ (${s eq 0}$) and the geodesic logarithmic potential $log(1/d)$. By analytic continuation we deduce the expansion for all complex values of $s$.