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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Optimal discrete measures for Riesz potentials


Authors: S. V. Borodachov, D. P. Hardin, A. Reznikov and E. B. Saff
Journal: Trans. Amer. Math. Soc. 370 (2018), 6973-6993
MSC (2010): Primary 31C20, 31C45; Secondary 28A78
DOI: https://doi.org/10.1090/tran/7224
Published electronically: April 17, 2018
MathSciNet review: 3841839
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Abstract: For weighted Riesz potentials of the form $K(x,y)=w(x,y)/$ $|x-y|^s$, we investigate $N$-point configurations $x_1,x_2, \ldots , x_N$ on a $d$-dimensional compact subset $A$ of $\mathbb {R}^p$ for which the minimum of $\sum _{j=1}^NK(x,x_j)$ on $A$ is maximal. Such quantities are called $N$-point Riesz $s$-polarization (or Chebyshev) constants. For $s\geqslant d$, we obtain the dominant term as $N\to \infty$ of such constants for a class of $d$-rectifiable subsets of $\mathbb {R}^p$. This class includes compact subsets of $d$-dimensional $C^1$ manifolds whose boundary relative to the manifold has $d$-dimensional Hausdorff measure zero, as well as finite unions of such sets when their pairwise intersections have measure zero. We also explicitly determine the weak-star limit distribution of asymptotically optimal $N$-point configurations for weighted $s$-polarization as $N\to \infty$.


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Additional Information

S. V. Borodachov
Affiliation: Department of Mathematics, Towson University, Towson, Maryland 21252
MR Author ID: 656604
Email: sborodachov@towson.edu

D. P. Hardin
Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37235
MR Author ID: 81245
ORCID: 0000-0003-0867-2146
Email: doug.hardin@vanderbilt.edu

A. Reznikov
Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37235
Address at time of publication: Department of Mathematics, Florida State University, Tallahassee, Florida 32306
MR Author ID: 895080
Email: reznikov@math.fsu.edu

E. B. Saff
Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37235
MR Author ID: 152845
Email: edward.b.saff@vanderbilt.edu

Received by editor(s): October 13, 2016
Received by editor(s) in revised form: February 6, 2017
Published electronically: April 17, 2018
Additional Notes: This research was supported, in part, by the U.S. National Science Foundation under the grant DMS-1412428 and DMS-1516400
Article copyright: © Copyright 2018 American Mathematical Society