Homogeneous extremal function for a ball in ℝ²

Mirosław Baran

Mirosław Baran

Annales Polonici Mathematici, Tome 72 (1999), p. 141-150 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Publié le : 1999-01-01

Zbl 0982.32028

EUDML-ID : urn:eudml:doc:262867

@article{bwmeta1.element.bwnjournal-article-apmv71z2p141bwm,
     author = {Miros\l aw Baran},
     title = {Homogeneous extremal function for a ball in $\mathbb{R}$$^2$},
     journal = {Annales Polonici Mathematici},
     volume = {72},
     year = {1999},
     pages = {141-150},
     zbl = {0982.32028},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv71z2p141bwm}
}

Mirosław Baran. Homogeneous extremal function for a ball in ℝ². Annales Polonici Mathematici, Tome 72 (1999) pp. 141-150. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv71z2p141bwm/

Bibliographie

[000] [B1] M. Baran, Siciak’s extremal function of convex sets in $ℂ^{n}$ , Ann. Polon. Math. 48 (1988), 275-280. | Zbl 0661.32023

[001] [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. | Zbl 0783.32009

[002] [B3] M. Baran, Complex equilibrium measure and Bernstein type theorems for compact sets in $ℝ^{n}$ , Proc. Amer. Math. Soc. 123 (1995), 485-494. | Zbl 0813.32011

[003] [B4] M. Baran, Bernstein type theorems for compact sets in $ℝ^{n}$ revisited, J. Approx. Theory 69 (1992), 156-166.

[004] [CL] E. W. Cheney and W. A. Light, Approximation Theory in Tensor Product Spaces, Lecture Notes in Math. 1169, Springer, Berlin, 1985. | Zbl 0575.41001

[005] [D] L. M. Drużkowski, Effective formula for the crossnorm in complexified unitary spaces, Univ. Iagel. Acta Math. 16 (1974), 47-53. | Zbl 0289.46017

[006] [Kl] M. Klimek, Pluripotential Theory, Oxford Univ. Press, 1991.

[007] [L] F. Leja, Teoria funkcji analitycznych [Theory of Analytic Functions], PWN, War- szawa, 1957 (in Polish).

[008] [Si1] J. Siciak, On an extremal function and domains of convergence of series of homogeneous polynomials, Ann. Polon. Math. 25 (1961), 297-307. | Zbl 0192.18102

[009] [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. | Zbl 0111.08102

[010] [Si3] J. Siciak, Holomorphic continuation of harmonic functions, Ann. Polon. Math. 29 (1974), 67-73. | Zbl 0247.32011

[011] [Si4] J. Siciak, Extremal plurisubharmonic functions in $ℂ^{n}$ , ibid. 39 (1981), 175-211.

[012] [Si5] J. Siciak, Extremal Plurisubharmonic Functions and Capacities in $ℂ^{n}$ , Sophia Kokyuroku in Math. 14, Sophia Univ., Tokyo, 1982.

[013] [St] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970.

[014] [SW] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971.