Homogeneous extremal function for a ball in ℝ²

Mirosław Baran

Annales Polonici Mathematici (1999)

  • Volume: 71, Issue: 2, page 141-150
  • ISSN: 0066-2216

Abstract

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We point out relations between Siciak’s homogeneous extremal function Ψ B and the Cauchy-Poisson transform in case B is a ball in ℝ². In particular, we find effective formulas for Ψ B for an important class of balls. These formulas imply that, in general, Ψ B is not a norm in ℂ².

How to cite

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Mirosław Baran. "Homogeneous extremal function for a ball in ℝ²." Annales Polonici Mathematici 71.2 (1999): 141-150. <http://eudml.org/doc/262867>.

@article{MirosławBaran1999,
abstract = {We point out relations between Siciak’s homogeneous extremal function $Ψ_B$ and the Cauchy-Poisson transform in case $B$ is a ball in ℝ². In particular, we find effective formulas for $Ψ_B$ for an important class of balls. These formulas imply that, in general, $Ψ_B$ is not a norm in ℂ².},
author = {Mirosław Baran},
journal = {Annales Polonici Mathematici},
keywords = {homogeneous extremal function; Cauchy-Poisson transform; plurisubharmonic functions},
language = {eng},
number = {2},
pages = {141-150},
title = {Homogeneous extremal function for a ball in ℝ²},
url = {http://eudml.org/doc/262867},
volume = {71},
year = {1999},
}

TY - JOUR
AU - Mirosław Baran
TI - Homogeneous extremal function for a ball in ℝ²
JO - Annales Polonici Mathematici
PY - 1999
VL - 71
IS - 2
SP - 141
EP - 150
AB - We point out relations between Siciak’s homogeneous extremal function $Ψ_B$ and the Cauchy-Poisson transform in case $B$ is a ball in ℝ². In particular, we find effective formulas for $Ψ_B$ for an important class of balls. These formulas imply that, in general, $Ψ_B$ is not a norm in ℂ².
LA - eng
KW - homogeneous extremal function; Cauchy-Poisson transform; plurisubharmonic functions
UR - http://eudml.org/doc/262867
ER -

References

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  1. [B1] M. Baran, Siciak’s extremal function of convex sets in ℂ n , Ann. Polon. Math. 48 (1988), 275-280. Zbl0661.32023
  2. [B2] M. Baran, Plurisubharmonic extremal functions and complex foliations for the complement of convex sets in ℝ n , Michigan Math. J. 39 (1992), 395-404. Zbl0783.32009
  3. [B3] M. Baran, Complex equilibrium measure and Bernstein type theorems for compact sets in ℝ n , Proc. Amer. Math. Soc. 123 (1995), 485-494. Zbl0813.32011
  4. [B4] M. Baran, Bernstein type theorems for compact sets in ℝ n revisited, J. Approx. Theory 69 (1992), 156-166. 
  5. [CL] E. W. Cheney and W. A. Light, Approximation Theory in Tensor Product Spaces, Lecture Notes in Math. 1169, Springer, Berlin, 1985. Zbl0575.41001
  6. [D] L. M. Drużkowski, Effective formula for the crossnorm in complexified unitary spaces, Univ. Iagel. Acta Math. 16 (1974), 47-53. Zbl0289.46017
  7. [Kl] M. Klimek, Pluripotential Theory, Oxford Univ. Press, 1991. 
  8. [L] F. Leja, Teoria funkcji analitycznych [Theory of Analytic Functions], PWN, War- szawa, 1957 (in Polish). 
  9. [Si1] J. Siciak, On an extremal function and domains of convergence of series of homogeneous polynomials, Ann. Polon. Math. 25 (1961), 297-307. Zbl0192.18102
  10. [Si2] J. Siciak, On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc. 105 (1962), 322-357. Zbl0111.08102
  11. [Si3] J. Siciak, Holomorphic continuation of harmonic functions, Ann. Polon. Math. 29 (1974), 67-73. Zbl0247.32011
  12. [Si4] J. Siciak, Extremal plurisubharmonic functions in ℂ n , ibid. 39 (1981), 175-211. 
  13. [Si5] J. Siciak, Extremal Plurisubharmonic Functions and Capacities in ℂ n , Sophia Kokyuroku in Math. 14, Sophia Univ., Tokyo, 1982. 
  14. [St] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970. 
  15. [SW] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971. 

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