Geodesic distance Riesz energy on the sphere

Bilyk, Dmitriy; Dai, Feng

doi:10.1090/tran/7711

Geodesic distance Riesz energy on the sphere
HTML articles powered by AMS MathViewer

by Dmitriy Bilyk and Feng Dai PDF

Trans. Amer. Math. Soc. 372 (2019), 3141-3166 Request permission

Abstract:

We study energy integrals and discrete energies on the sphere, in particular, analogues of the Riesz energy with the geodesic distance in place of the Euclidean, and we determine that the range of exponents for which uniform distribution optimizes such energies is different from the classical case. We also obtain a very general form of the Stolarsky principle, which relates discrete energies to certain $L^2$ discrepancies, and prove optimal asymptotic estimates for both objects. This leads to sharp asymptotics of the difference between optimal discrete and continuous energies in the geodesic case, as well as new proofs of discrepancy estimates.

References

George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958, DOI 10.1017/CBO9781107325937
József Beck, Sums of distances between points on a sphere—an application of the theory of irregularities of distribution to discrete geometry, Mathematika 31 (1984), no. 1, 33–41. MR 762175, DOI 10.1112/S0025579300010639
Aline Bonami and Jean-Louis Clerc, Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques, Trans. Amer. Math. Soc. 183 (1973), 223–263 (French). MR 338697, DOI 10.1090/S0002-9947-1973-0338697-5
Andriy Bondarenko, Danylo Radchenko, and Maryna Viazovska, Optimal asymptotic bounds for spherical designs, Ann. of Math. (2) 178 (2013), no. 2, 443–452. MR 3071504, DOI 10.4007/annals.2013.178.2.2
Dmitriy Bilyk, Feng Dai, and Ryan Matzke, The Stolarsky principle and energy optimization on the sphere, Constr. Approx. 48 (2018), no. 1, 31–60. MR 3825946, DOI 10.1007/s00365-017-9412-4
D. Bilyk, F. Dai, and S. Steinerberger, General and refined Montgomery lemmata, https://arxiv.org/abs/1801.07701 (2018).
Dmitriĭ Bilik and Maĭkl T. Lèĭsi, One bit sensing, discrepancy, and the Stolarsky principle, Mat. Sb. 208 (2017), no. 6, 4–25 (Russian, with Russian summary); English transl., Sb. Math. 208 (2017), no. 5-6, 744–763. MR 3659577, DOI 10.4213/sm8656
Göran Björck, Distributions of positive mass, which maximize a certain generalized energy integral, Ark. Mat. 3 (1956), 255–269. MR 78470, DOI 10.1007/BF02589412
Johann S. Brauchart, Invariance principles for energy functionals on spheres, Monatsh. Math. 141 (2004), no. 2, 101–117. MR 2037987, DOI 10.1007/s00605-002-0007-0
Johann S. Brauchart, About the second term of the asymptotics for optimal Riesz energy on the sphere in the potential-theoretical case, Integral Transforms Spec. Funct. 17 (2006), no. 5, 321–328. MR 2237493, DOI 10.1080/10652460500431859
Johann S. Brauchart and Josef Dick, A simple proof of Stolarsky’s invariance principle, Proc. Amer. Math. Soc. 141 (2013), no. 6, 2085–2096. MR 3034434, DOI 10.1090/S0002-9939-2013-11490-5
J. S. Brauchart, D. P. Hardin, and E. B. Saff, The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere, Recent advances in orthogonal polynomials, special functions, and their applications, Contemp. Math., vol. 578, Amer. Math. Soc., Providence, RI, 2012, pp. 31–61. MR 2964138, DOI 10.1090/conm/578/11483
Johann S. Brauchart, Douglas P. Hardin, and Edward B. Saff, Discrete energy asymptotics on a Riemannian circle, Unif. Distrib. Theory 7 (2012), no. 2, 77–108. MR 3052946
Feng Dai and Yuan Xu, Approximation theory and harmonic analysis on spheres and balls, Springer Monographs in Mathematics, Springer, New York, 2013. MR 3060033, DOI 10.1007/978-1-4614-6660-4
L. Fejes Tóth, Über eine Punktverteilung auf der Kugel, Acta Math. Acad. Sci. Hungar. 10 (1959), 13–19 (unbound insert) (German, with Russian summary). MR 105654, DOI 10.1007/BF02063286
C. L. Frenzen and R. Wong, A uniform asymptotic expansion of the Jacobi polynomials with error bounds, Canad. J. Math. 37 (1985), no. 5, 979–1007. MR 806651, DOI 10.4153/CJM-1985-053-5
George Gasper, Positive integrals of Bessel functions, SIAM J. Math. Anal. 6 (1975), no. 5, 868–881. MR 390318, DOI 10.1137/0506076
Yitzhak Katznelson, An introduction to harmonic analysis, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. MR 2039503, DOI 10.1017/CBO9781139165372
A. B. J. Kuijlaars and E. B. Saff, Asymptotics for minimal discrete energy on the sphere, Trans. Amer. Math. Soc. 350 (1998), no. 2, 523–538. MR 1458327, DOI 10.1090/S0002-9947-98-02119-9
Heinrich Larcher, Solution of a gemetric problem by Fejes Tòth, Michigan Math. J. 9 (1962), 45–51. MR 137032
Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1297543, DOI 10.1090/cbms/084
H. He, K. Basu, Q. Zhao, and A. Owen, Permutation $p$-value approximation via generalized Stolarsky invariance, https://arxiv.org/abs/1603.02757 (2016).
E. A. Rakhmanov, E. B. Saff, and Y. M. Zhou, Minimal discrete energy on the sphere, Math. Res. Lett. 1 (1994), no. 6, 647–662. MR 1306011, DOI 10.4310/MRL.1994.v1.n6.a3
I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J. 9 (1942), 96–108. MR 5922
Richard Evan Schwartz, The five-electron case of Thomson’s problem, Exp. Math. 22 (2013), no. 2, 157–186. MR 3047910, DOI 10.1080/10586458.2013.766570
M. M. Skriganov, Point distributions in compact metric spaces, Mathematika 63 (2017), no. 3, 1152–1171. MR 3731319, DOI 10.1112/S0025579317000286
M. Skriganov, Point distributions in compact metric spaces, II, http://www.pdmi.ras.ru/preprint/2016/16-02.html (2016).
M. Skriganov, Point distributions in compact metric spaces, III, http://www.pdmi.ras.ru/preprint/2016/16-07.html (2016).
Günter Sperling, Lösung einer elementargeometrischen Frage von Fejes Tóth, Arch. Math. 11 (1960), 69–71 (German). MR 112077, DOI 10.1007/BF01236910
Stefan Steinerberger, Exponential sums and Riesz energies, J. Number Theory 182 (2018), 37–56. MR 3703931, DOI 10.1016/j.jnt.2017.08.002
Kenneth B. Stolarsky, Sums of distances between points on a sphere. II, Proc. Amer. Math. Soc. 41 (1973), 575–582. MR 333995, DOI 10.1090/S0002-9939-1973-0333995-9
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
Yan Shuo Tan, Energy optimization for distributions on the sphere and improvement to the Welch bounds, Electron. Commun. Probab. 22 (2017), Paper No. 43, 12. MR 3693769, DOI 10.1214/17-ECP73
Gerold Wagner, On means of distances on the surface of a sphere (lower bounds), Pacific J. Math. 144 (1990), no. 2, 389–398. MR 1061328
Gerold Wagner, On means of distances on the surface of a sphere. II. Upper bounds, Pacific J. Math. 154 (1992), no. 2, 381–396. MR 1159518