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ASYMPTOTIC LINEAR PROGRAMMING LOWER BOUNDS FOR THE ENERGY OF MINIMIZING RIESZ AND GAUSS CONFIGURATIONS

Part of: Classical measure theory Other generalizations

Published online by Cambridge University Press:  27 September 2018

D. P. Hardin ,
T. J. Michaels  and
E. B. Saff
D. P. Hardin
Affiliation:
Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, TN 37240, U.S.A. email doug.hardin@vanderbilt.edu
T. J. Michaels
Affiliation:
Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, TN 37240, U.S.A. email tim.michaels12@gmail.com
E. B. Saff
Affiliation:
Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, TN 37240, U.S.A. email edward.b.saff@vanderbilt.edu
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Abstract

Utilizing frameworks developed by Delsarte, Yudin and Levenshtein, we deduce linear programming lower bounds (as $N\rightarrow \infty$) for the Riesz energy of $N$-point configurations on the $d$-dimensional unit sphere in the so-called hypersingular case; i.e., for non-integrable Riesz kernels of the form $|x-y|^{-s}$ with $s>d$. As a consequence, we immediately get (thanks to the poppy-seed bagel theorem) lower estimates for the large $N$ limits of minimal hypersingular Riesz energy on compact $d$-rectifiable sets. Furthermore, for the Gaussian potential $\exp (-\unicode[STIX]{x1D6FC}|x-y|^{2})$ on $\mathbb{R}^{p}$, we obtain lower bounds for the energy of infinite configurations having a prescribed density.

MSC classification

Primary: 28A78: Hausdorff and packing measures 31C20: Discrete potential theory and numerical methods
Type
Research Article
Information
Mathematika , Volume 65 , Issue 1 , 2019 , pp. 157 - 180
Copyright
Copyright © University College London 2018 

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Footnotes

The research of the authors was supported, in part, by National Science Foundation grant DMS-1516400. The research of T. Michaels was completed as part of his PhD dissertation at Vanderbilt University. Research for this article was conducted while two of the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the “Point Configurations in Geometry, Physics and Computer Science” program supported by the National Science Foundation under Grant No. DMS-1439786 and by a Simons Foundation Targeted Grant.

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