LeVeque type inequalities and discrepancy estimates for minimal energy configurations on spheres

Let S^d denote the unit sphere in the Euclidean space R^d^+^1(d>=1). We develop LeVeque type inequalities for the discrepancy between the rotationally invariant probability measure and the normalized counting measures on S^d. We obtain both upper bound and lower bound estimates. We then use these inequalities to estimate the discrepancy of the normalized counting measures associated with minimal energy configurations on S^d.

[1]  Gerold Wagner,et al.  On means of distances on the surface of a sphere. II. (Upper bounds) , 1990 .

[2]  Peter J. Grabner,et al.  Erdös-Turán type discrepancy bounds , 1991 .

[3]  Xingping Sun,et al.  Approximation power of RBFs and their associated SBFs: a connection , 2007, Adv. Comput. Math..

[4]  Frank Filbir,et al.  Radial basis functions and corresponding zonal series expansions on the sphere , 2005, J. Approx. Theory.

[5]  G. A. Watson.  A treatise on the theory of Bessel functions , 1944 .

[6]  J. Beck.  Sums of distances between points on a sphere — an application of the theory of irregularities of distribution to discrete Geometry , 1984 .

[7]  M. Wodzicki.  Lecture Notes in Math , 1984 .

[8]  Xingping Sun,et al.  Strictly positive definite functions on spheres in Euclidean spaces , 1996, Math. Comput..

[9]  R. Alexander,et al.  On the sum of distances betweenn points on a sphere , 1972 .

[10]  Joseph D. Ward,et al.  Scattered Data Interpolation on Spheres: Error Estimates and Locally Supported Basis Functions , 2002, SIAM J. Math. Anal..

[11]  Robert F. Tichy,et al.  Sequences, Discrepancies and Applications , 1997 .

[12]  Steven A. Orszag,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 1978 .

[13]  Hermann Weyl.  Über die Gleichverteilung von Zahlen , 1928 .

[14]  Volker Schönefeld.  Spherical Harmonics , 2019, An Introduction to Radio Astronomy.

[15]  E. Saff,et al.  Asymptotics for minimal discrete energy on the sphere , 1995 .

[16]  I. J. Schoenberg.  Metric spaces and completely monotone functions , 1938 .

[17]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[18]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[19]  Lauwerens Kuipers,et al.  Uniform distribution of sequences , 1974 .

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

[21]  J. Vaaler.  SOME EXTREMAL FUNCTIONS IN FOURIER ANALYSIS , 2007 .

[22]  H. Weyl.  Über die Gleichverteilung von Zahlen mod. Eins , 1916 .

[23]  William W. L. Chen.  On irregularities of distribution. , 1980 .

[24]  R. Alexander,et al.  On the sum of distances betweenn points on a sphere. II , 1972 .

[25]  J. Vaaler,et al.  SOME TRIGONOMETRIC EXTREMAL FUNCTIONS AND THE ERDOS-TURAN TYPE INEQUALITIES , 1999 .

[26]  E. Wagner.  International Series of Numerical Mathematics , 1963 .

[27]  G. Pólya,et al.  Über den transfiniten Durchmesser (Kapazitätskonstante) von ebenen und räumlichen Punktmengen. , 1931 .

[28]  E. Haacke.  Sequences , 2005 .

[29]  Gerold Wagner.  On a new method for constructing good point sets on spheres , 1993, Discret. Comput. Geom..

[30]  F. J. Narcowich,et al.  Variational Principles and Sobolev-Type Estimates for Generalized Interpolation on a Riemannian Manifold , 1999 .

[31]  Harald Niederreiter,et al.  Monte-Carlo and Quasi-Monte Carlo Methods 1998 , 2000 .

[32]  J. Brauchart.  Note on a Generalized Invariance Principle and its Relevance for Cap Discrepancy and Energy , 2003 .

[33]  I. J. Schoenberg.  Positive definite functions on spheres , 1942 .

[34]  An inequality connected with Weyl''s criterion for uniform distribution , 1965 .

[35]  F. Su.  A LeVeque-type lower bound for discrepancy , 2000 .

[36]  GEROLD WAGNER,et al.  ON MEANS OF DISTANCES ON THE SURFACE OF A SPHERE (LOWER BOUNDS) , 2012 .

[37]  Scattered data interpolation by linear combinations of translates of conditionally positive definite functions , 1991 .

[38]  H. Montgomery.  Ten lectures on the interface between analytic number theory and harmonic analysis , 1994 .

[39]  K. Stolarsky.  Sums of distances between points on a sphere. II , 1972 .

[40]  E. Saff,et al.  Minimal Riesz Energy Point Configurations for Rectifiable d-Dimensional Manifolds , 2003, math-ph/0311024.

[41]  Johann S. Brauchart,et al.  Optimal logarithmic energy points on the unit sphere , 2008, Math. Comput..