On the multivariate transfinite diameter

• Autorzy: Thomas Bloom , Jean-Paul Calvi
• Języki publikacji: EN

Abstrakty

EN

We prove several new results on the multivariate transfinite diameter and its connection with pluripotential theory: a formula for the transfinite diameter of a general product set, a comparison theorem and a new expression involving Robin's functions. We also study the transfinite diameter of the pre-image under certain proper polynomial mappings.

Słowa kluczowe

EN

Robin's functions   extremal plurisubharmonic functions   minimal polynomials   multivariate transfinite diameter   Chebyshev polynomials  

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1999
• Tom: 72
• Numer: 3
• Strony: 285-305

Daty

wydano

1999

otrzymano

1999-06-25

poprawiono

1999-10-11

poprawiono

1999-11-10

Twórcy

autor

Thomas Bloom

• Department of Mathematics, University of Toronto, M5S 3G3, Toronto, Ontario, Canada

autor

Jean-Paul Calvi

• Laboratoire de Mathématiques E. Picard, UFR MIG, Université Paul Sabatier, 31062 Toulouse Cedex, France

Bibliografia

• [Ba] M. Baran, Conjugate norms in $ℂ^n$ and related geometrical problems, Dissertationes Math. 378 (1998).
• [BT] E. Bedford and B. A. Taylor, Plurisubharmonic functions with logarithmic singularities, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 4, 133-171.
• [BT2] E. Bedford and B. A. Taylor, Fine topology, Šilov boundary, and $(dd^c)^n$, J. Funct. Anal. 72 (1987), 225-251.
• [Blo] Z. Błocki, The equilibrium measure for product subsets of $ℂ^n$, preprint, 1999.
• [Bl] T. Bloom, Some applications of the Robin function to multivariable approximation theory, J. Approx. Theory 92 (1998), 1-21.
• [BBCL] T. Bloom, L. Bos, C. Christensen and N. Levenberg, Polynomial interpolation of holomorphic functions in ℂ and $ℂ^n$, Rocky Mountain J. Math. 22 (1992), 441-470.
• [BC] T. Bloom and J.-P. Calvi, On multivariate minimal polynomials, preprint, 1999.
• [Bo] L. Bos, A characteristic of points in $ℝ^2$ having a Lebesgue function of exponential growth, J. Approx. Theory 56 (1989), 316-329.
• [Go] G. M. Goluzin, Geometric Theory of a Function of a Complex Variable, Amer. Math. Soc., Providence, 1969.
• [Je] M. Jędrzejowski, The homogeneous transfinite diameter of a compact subset of $ℂ^n$, Ann. Polon. Math. 55 (1991), 191-205.
• [Kl] M. Klimek, Pluripotential Theory, Oxford Univ. Press, Oxford, 1991.
• [Kl2] M. Klimek, Metrics associated with extremal plurisubharmonic functions, Proc. Amer. Math. Soc. 125 (1995), 2763-2770.
• [Le] N. Levenberg, Capacities in Several Complex Variables, thesis, University of Michigan, 1984.
• [LT] N. Levenberg and B. A. Taylor, Comparison of capacities in $ℂ^n$, in: Lectures Notes in Math. 1094, Springer, 1984, 162-172.
• [NZ] T. V. Nguyen et A. Zeriahi, Familles de polynômes presque partout bornées, Bull. Sci. Math. (2) 107 (1983), 81-91.
• [Ra] T. Ransford, Potential Theory in the Complex Plane, Cambridge Univ. Press, Cambridge, 1995.
• [SS] M. Schiffer and J. Siciak, Transfinite diameter and analytic continuation of functions of two complex variables, in: Studies in Math. Analysis and Related Topics, Stanford Univ. Press, 1962, 341-358.
• [Sh] V. P. Sheĭnov, Invariant form of Pólya's inequalities, Siberian Math. J. 14 (1973), 138-145.
• [Si1] J. Siciak, Extremal plurisubharmonic functions on $ℂ^N$, Ann. Polon. Math. 39 (1981), 175-211.
• [Si2] J. Siciak, A remark on Tchebysheff polynomials in $ℂ^n$, Univ. Iagell. Acta Math. 35 (1997), 37-45.
• [Za] V. P. Zaharjuta [V. P. Zakharyuta], Transfinite diameter, Chebyshev constants, and capacity for compacta in $ℂ^n$, Math. USSR-Sb. 25 (1975), 350-364.