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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Optimal discrete measures for Riesz potentials
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by S. V. Borodachov, D. P. Hardin, A. Reznikov and E. B. Saff PDF
Trans. Amer. Math. Soc. 370 (2018), 6973-6993 Request permission

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
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 6973-6993
  • MSC (2010): Primary 31C20, 31C45; Secondary 28A78
  • DOI: https://doi.org/10.1090/tran/7224
  • MathSciNet review: 3841839