Arkiv för Matematik

Transfinite diameter notions in ℂN and integrals of Vandermonde determinants

Thomas Bloom and Norman Levenberg

Full-text: Open access

Abstract

We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite diameter, and weighted transfinite diameter for sets in ℂN. An ingredient is a formula of Rumely (A Robin formula for the Fekete–Leja transfinite diameter, Math. Ann. 337 (2007), 729–738) which relates the Robin function and the transfinite diameter of a compact set. We also prove limiting formulas for integrals of generalized Vandermonde determinants with varying weights for a general class of compact sets and measures in ℂN. Our results extend to certain weights and measures defined on cones in ℝN.

Note

Bloom supported in part by an NSERC grant.

Article information

Source
Ark. Mat., Volume 48, Number 1 (2010), 17-40.

Dates
Received: 7 February 2008
Revised: 15 March 2009
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907096

Digital Object Identifier
doi:10.1007/s11512-009-0101-9

Mathematical Reviews number (MathSciNet)
MR2594584

Zentralblatt MATH identifier
1196.31003

Rights
2009 © Institut Mittag-Leffler

Citation

Bloom, Thomas; Levenberg, Norman. Transfinite diameter notions in ℂ N and integrals of Vandermonde determinants. Ark. Mat. 48 (2010), no. 1, 17--40. doi:10.1007/s11512-009-0101-9. https://projecteuclid.org/euclid.afm/1485907096


Export citation

References

  • Berman, R. and Boucksom, S., Growth of balls of holomorphic sections and energy at equilibrium, Preprint.
  • Berman, R. and Boucksom, S., Ball volume ratios, energy functionals and transfinite diameter for line bundles, Preprint.
  • Berman, R. and Boucksom, S., Capacities and weighted volumes of line bundles, Preprint, 2008.
  • Berman, R. and Boucksom, S., Equidistribution of Fekete points on complex manifolds, Preprint, 2008.
  • Berman, R., Boucksom, S. and Nystrom, D. W., Convergence towards equilibrium on complex manifolds, Preprint.
  • Berman, R. and Nystrom, D. W., Convergence of Bergman measures for high powers of a line bundle, Preprint, 2008.
  • Bloom, T., Some applications of the Robin function to multivariable approximation theory, J. Approx. Theory 92 (1998), 1–21.
  • Bloom, T., Weighted polynomials and weighted pluripotential theory, Trans. Amer. Math. Soc. 361 (2009), 2163–2179.
  • Bloom, T., Bos, L., Christensen, C. and Levenberg, N., Polynomial interpolation of holomorphic functions in ℂ and ℂn, Rocky Mountain J. Math. 22 (1992), 441–470.
  • Bloom, T. and Levenberg, N., Capacity convergence results and applications to a Bernstein–Markov inequality, Trans. Amer. Math. Soc. 351 (1999), 4753–4767.
  • Bloom, T. and Levenberg, N., Weighted pluripotential theory in ℂN, Amer. J. Math. 125 (2003), 57–103.
  • Bloom, T. and Levenberg, N., Asymptotics for Christoffel functions of planar measures, J. Anal. Math. 106 (2008), 353–371.
  • Bloom, T., Levenberg, N. and Ma’u, S., Robin functions and extremal functions, Ann. Polon. Math. 80 (2003), 55–84.
  • Deift, P., Orthogonal Polynomials and Random Matrices : A Riemann–Hilbert Approach, Amer. Math. Soc., Providence, RI, 1999.
  • Jedrzejowski, M., The homogeneous transfinite diameter of a compact subset of ℂN, Ann. Polon. Math. 55 (1991), 191–205.
  • Johansson, K., On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 91 (1998), 151–204.
  • Klimek, M., Pluripotential Theory, Oxford University Press, New York, 1991.
  • Mehta, M. L., Random Matrices, 3rd ed., Pure Appl. Math. 142, Elsevier/Academic Press, Amsterdam, 2004.
  • Rumely, R., A Robin formula for the Fekete–Leja transfinite diameter, Math. Ann. 337 (2007), 729–738.
  • Saff, E. and Totik, V., Logarithmic Potentials with External Fields, Springer, Berlin, 1997.
  • Siciak, J., Extremal plurisubharmonic functions in ℂn, Ann. Polon. Math. 39 (1981), 175–211.
  • Zaharjuta, V. P., Transfinite diameter, Chebyshev constants, and capacity for compacta in ℂn, Mat. Sb. 96 (138) (1975), 374–389, 503 (Russian). English transl.: Math. USSR-Sb. 25 (1975), 350–364.