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 in Rp, p � 2, as N ! 1 . For f decreasing and convex, such a point configuration minimizes the f-energy P j6k f(d(xj ,xk)), where d is the geodesic distance (with respect to ) between points on . Co mpletely 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 6 0) and the geodesic logarithmic potential log(1/d). By analytic continuation we deduce the expansion for all complex values of s. Communicated by

[1]  José L. López,et al.  Asymptotic Expansions of Mellin Convolution Integrals , 2008, SIAM Rev..

[2]  L. Fejes Tóth,et al.  On the sum of distances determined by a pointset , 1956 .

[3]  J. S. Lew,et al.  Asymptotic Expansion of Laplace Transforms Near the Origin , 1970 .

[4]  Ralph Alexander,et al.  Extremal problems of distance geometry related to energy integrals , 1974 .

[5]  M. J. Dubourdieu.  Sur un théorème de M. S. Bernstein relatif à la transformation de Laplace-Stieltjes , 1940 .

[6]  V. Maymeskul,et al.  Asymptotics for Minimal Discrete Riesz Energy on Curves in ℝ d , 2004, Canadian Journal of Mathematics.

[7]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[8]  Edward B. Saff,et al.  Fundamentals of complex analysis for mathematics, science, and engineering , 1976 .

[9]  N. S. Landkof.  Foundations of Modern Potential Theory , 1972 .

[10]  Avram Sidi,et al.  Euler–Maclaurin Expansions for Integrals with Arbitrary Algebraic-Logarithmic Endpoint Singularities , 2011, Constructive Approximation.

[11]  J. Brauchart,et al.  A remark on exact formulas for the Riesz energy of the $N$th roots of unity , 2011 .

[12]  D. Widder,et al.  The Laplace Transform , 1943, The Mathematical Gazette.

[13]  C. W. Clenshaw,et al.  The special functions and their approximations , 1972 .

[14]  Masanori Katsurada,et al.  Power series and asymptotic series associated with the Lerch zeta-function , 1998 .

[15]  E. Saff,et al.  The Riesz energy of the Nth roots of unity: an asymptotic expansion for large N , 2008, 0808.1291.

[16]  Henry Cohn,et al.  Universally optimal distribution of points on spheres , 2006, math/0607446.

[17]  L. Tóth,et al.  Über eine Punktverteilung auf der Kugel , 1959 .

[18]  Stamatis Koumandos,et al.  Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function , 2008, 0811.2870.

[19]  J. S. Lew,et al.  Asymptotic expansion of a class of integral transforms with algebraically dominated kernels , 1971 .

[20]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[21]  Editors , 1986, Brain Research Bulletin.

[22]  Mario Götz,et al.  On the Riesz energy of measures , 2003, J. Approx. Theory.

[23]  T. Apostol.  An Elementary View of Euler's Summation Formula , 1999 .

[24]  Günter Sperling.  Lösung einer elementargeometrischen Frage von FEJES TÒTH , 1960 .

[25]  E. Saff,et al.  Asymptotics for discrete weighted minimal Riesz energy problems on rectifiable sets , 2006, math-ph/0602025.

[26]  Asymptotic Expansion of Laplace Transforms about the Origin using Generalized Functions , 1979 .

[27]  G. Björck,et al.  Distributions of positive mass, which maximize a certain generalized energy integral , 1956 .

[28]  ASYMPTOTIC RESULTS FOR THE MINIMUM ENERGY AND BEST PACKING PROBLEMS ON RECTIFIABLE SETS By , 2006 .

[29]  Minghui Jiang,et al.  On the sum of distances along a circle , 2008, Discret. Math..

[30]  S. V. Borodachov,et al.  Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves , 2012, Canadian Journal of Mathematics.

[31]  Duane W. DeTemple,et al.  Half integer approximations for the partial sums of the harmonic series , 1991 .