Article
Discrete Energy Asymptotics on a Riemannian Circle
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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 Rp, p ≥ 2, as N → ∞. For f decreasing and convex, such a point configuration minimizes the f-energy P j6=k f(d(x j, x k)), where d is the ge- odesic distance (with respect to Γ) between points on Γ. Completely monotonic functions, analytic kernel functions, Laurent series, and weighted kernel func- tions f are studied. Of particular interest are the geodesic Riesz potential 1/ds (s 6= 0) and the geodesic logarithmic potential log(1/d). By analytic continuation we deduce the expansion for all complex values of s.
Keywords
Discrete Energy Asymptotics, Geodesic Riesz Energy, Geodesic Logarithmic Energy, Riemannian Circle, Riemann Zeta Function, General Kernel Functions, Euler-MacLaurin Summation Formula.
Citation
Brauchart, J. S., Hardin, D. P., & Saff, E. B. (2012). Discrete energy asymptotics on a riemannian circle. Uniform Distribution Theory, 7(2), 77–108.
J. S. Brauchart, D. P. Hardin and E. B. Saff, “Discrete energy asymptotics on a riemannian circle,” Uniform Distribution Theory, vol. 7, no. 2, pp. 77–108, 2012.
Brauchart JS, Hardin DP, Saff EB. Discrete energy asymptotics on a riemannian circle. Uniform Distribution Theory. 2012;7(2):77–108.
Brauchart, J. S., Hardin, D. P. and Saff, E. B. (2012), ‘Discrete energy asymptotics on a riemannian circle’, Uniform Distribution Theory, 7(2), pp. 77–108.
Brauchart, J. S., et al. “Discrete Energy Asymptotics on a Riemannian Circle.” Uniform Distribution Theory, vol. 7, no. 2, 2012, pp. 77–108.
Brauchart, J. S., D. P. Hardin, and E. B. Saff. “Discrete Energy Asymptotics on a Riemannian Circle.” Uniform Distribution Theory 7, no. 2 (2012): 77–108.
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Published by: Engineering Journals
