Complete minimal logarithmic energy asymptotics for points in a compact interval: a consequence of the discriminant of Jacobi polynomials

The electrostatic interpretation of zeros of Jacobi polynomials, due to Stieltjes and Schur, enables us to obtain the complete asymptotic expansion as n → ∞ of the minimal logarithmic potential energy of n point charges restricted to move in the interval [−1, 1] in the presence of an external field generated by endpoint charges. By the same methods, we determine the complete asymptotic expansion of the logarithmic energy ∑ j 6=k log(1/|xj − xk|) of Fekete points, which, by definition, maximize the product of all mutual distances ∏ j 6=k |xj − xk| of N points in [−1, 1] as N → ∞. The results for other compact intervals differ only in the quadratic and linear term of the asymptotics. Explicit formulas and their asymptotics follow from the discriminant, leading coefficient, and special values at ±1 of Jacobi polynomials. For all these quantities we derive complete Poincaré-type asymptotics.

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

[2]  Zur approximation des transfiniten durchmessers bei bis auf ecken analytischen geschlossenen jordankurven , 1974 .

[3]  Über die Faberschen Polynome schlichter Funktionen , 1964 .

[4]  E. Saff,et al.  The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere , 2012, 1202.4037.

[5]  Masato Kimura,et al.  Quantitative Estimate of the Continuum Approximations of Interacting Particle Systems in One Dimension , 2021, SIAM J. Math. Anal..

[6]  S. Serfaty,et al.  Crystallization for Coulomb and Riesz interactions as a consequence of the Cohn-Kumar conjecture , 2019, Proceedings of the American Mathematical Society.

[7]  S. Serfaty,et al.  2D Coulomb Gases and the Renormalized Energy , 2012, 1201.3503.

[8]  Explicit formulas for the Riesz energy of the $N$th roots of unity , 2011, 1105.5530.

[9]  P. Dragnev,et al.  On point-mass Riesz external fields on the real axis , 2019, Journal of Mathematical Analysis and Applications.

[10]  M. Skriganov.  POINT DISTRIBUTIONS IN TWO‐POINT HOMOGENEOUS SPACES , 2019, Mathematika.

[11]  Victor S. Adamchik,et al.  Polygamma functions of negative order , 1998 .

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

[13]  Johann S. Brauchart,et al.  Menke points on the real line and their connection to classical orthogonal polynomials , 2010, J. Comput. Appl. Math..

[14]  Carlos Beltrán,et al.  The Diamond ensemble: A constructive set of spherical points with small logarithmic energy , 2020, J. Complex..

[15]  Johann S. Brauchart,et al.  Distributing many points on spheres: Minimal energy and designs , 2014, J. Complex..

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

[17]  S. Serfaty,et al.  Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals , 2013, 1307.2805.

[18]  Florian Pausinger.  Greedy energy minimization can count in binary: point charges and the van der Corput sequence , 2019, Annali di Matematica Pura ed Applicata (1923 -).

[19]  Mourad E. H. Ismail,et al.  Special functions, q-series, and related topics , 1997 .

[20]  Steven R. Finch,et al.  Mathematical constants , 2005, Encyclopedia of mathematics and its applications.

[21]  C. Beltr'an,et al.  A sequence of polynomials with optimal condition number , 2019, Journal of the American Mathematical Society.

[22]  Alexander Barg,et al.  Stolarsky's invariance principle for finite metric spaces , 2020, ArXiv.

[23]  J. Dick,et al.  A simple proof of Stolarsky’s invariance principle , 2011, 1101.4448.

[24]  E. Saff,et al.  Distributing many points on a sphere , 1997 .

[25]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[26]  Douglas P Hardin,et al.  Discrete Energy on Rectifiable Sets , 2019, Springer Monographs in Mathematics.

[27]  E. Saff,et al.  Discretizing Manifolds via Minimum Energy Points , 2004 .

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

[29]  Carlos Beltrán.  A Facility Location Formulation for Stable Polynomials and Elliptic Fekete Points , 2015, Found. Comput. Math..

[30]  E. Saff,et al.  DISCRETE ENERGY ASYMPTOTICS ON A RIEMANNIAN CIRCLE , 2009, 0912.4720.

[31]  Carlos Beltrán,et al.  Energy and discrepancy of rotationally invariant determinantal point processes in high dimensional spheres , 2015, J. Complex..

[32]  Edward B. Saff,et al.  Asymptotics of greedy energy points , 2009, Math. Comput..

[33]  D. A. Dunnett.  Classical Electrodynamics , 2020, Nature.

[34]  Etienne Sandier,et al.  Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere , 2014, 1404.4485.

[35]  Henry Cohn,et al.  The sphere packing problem in dimension 8The sphere packing problem in dimension 8 , 2016, 1603.04246.

[36]  Th. Clausen.  Über Interpolation. , 2009 .

[37]  Mourad E. H. Ismail,et al.  An electrostatics model for zeros of general orthogonal polynomials , 2000 .

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

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

[40]  Ryan Matzke,et al.  Energy on spheres and discreteness of minimizing measures , 2019, Journal of Functional Analysis.

[41]  Maya Stoyanova,et al.  Energy bounds for codes and designs in Hamming spaces , 2015, Designs, Codes and Cryptography.

[42]  Leo P. Kadanoff,et al.  Discrete Charges on a Two Dimensional Conductor , 2004 .

[43]  S. Serfaty,et al.  NEXT ORDER ASYMPTOTICS AND RENORMALIZED ENERGY FOR RIESZ INTERACTIONS , 2014, Journal of the Institute of Mathematics of Jussieu.

[44]  S. Serfaty,et al.  1D log gases and the renormalized energy: crystallization at vanishing temperature , 2013, 1303.2968.

[45]  Extremalpunkte und konforme Abbildung , 1971 .

[46]  F. Marcellán,et al.  Electrostatic models for zeros of polynomials: Old, new, and some open problems , 2005 .

[47]  Thomas Leblé.  Logarithmic, Coulomb and Riesz Energy of Point Processes , 2015, 1509.05253.