Advertisement

Potential Analysis

, Volume 47, Issue 2, pp 235–244 | Cite as

A Minimum Principle for Potentials with Application to Chebyshev Constants

  • A. ReznikovEmail author
  • E. B. Saff
  • O. V. Vlasiuk
Article

Abstract

For “Riesz-like” kernels K(x,y) = f(|xy|) on A×A, where A is a compact d-regular set \(A\subset \mathbb {R}^{p}\), we prove a minimum principle for potentials \(U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)\), where μ is a Borel measure supported on A. Setting \(P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)\), the K-polarization of μ, the principle is used to show that if {ν N } is a sequence of measures on A that converges in the weak-star sense to the measure ν, then P K (ν N )→P K (ν) as \(N\to \infty \). The continuous Chebyshev (polarization) problem concerns maximizing P K (μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes P K (μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {ν N } is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of ν N as \(N \to \infty \) is a solution to the continuous problem.

Keywords

Maximal Riesz polarization Chebyshev constant Hausdorff measure Riesz potential Minimum principle 

Mathematics Subject Classification (2010)

Primary 31C15, 31C20 Secondary 30C80 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Borodachov, S.V., Hardin, D.P., Saff, E.B.: Minimal discrete energy on rectifiable sets. Springer (2017). (to appear)Google Scholar
  2. 2.
    Brelot, M.: Lectures on potential theory available at www.math.tifr.res.in/publ/ln/tifr19.pdf/ (1960)
  3. 3.
    Erdélyi, T., Saff, E.B.: Riesz polarization inequalities in higher dimensions. J. Approx. Theory 171, 128–147 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Farkas, B., Nagy, B.: Transfinite diameter, Chebyshev constant and energy on locally compact spaces. Potential Anal. 28, 241–260 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Farkas, B., Révész, S.G.: Potential theoretic approach to rendezvous numbers. Monatsh. Math. 148(4), 309–331 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Farkas, B., Révész, S. G.: Rendezvous numbers of metric spaces—a potential theoretic approach. Arch. Math. (Basel) 86(3), 268–281 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fuglede, B.: On the theory of potentials in locally compact spaces. Acta Math. 103, 139–215 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ohtsuka, M.: On various definitions of capacity and related notions. Nagoya Math. J. 30, 121–127 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Simanek, B.: Extremal polarization configurations for integrable kernels. New York J. Math. 22, 667–675 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Center for Constructive Approximation, Department of MathematicsVanderbilt UniversityNashvilleUSA

Personalised recommendations