On some extremal functions and their applications in the theory of analytic functions of several complex variables

Sičiak, Józef

doi:10.1090/s0002-9947-1962-0143946-5

Cited by 144 publications

(76 citation statements)

References 14 publications

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“…Recall that the extremal function, associated with a (nonempty) compact set K ⊂ C N and introduced by J. Siciak in [26], is defined by the formula K (z) := sup{| p(z)| 1/deg p : p ∈ C[Z ] is nonconstant and p K ≤ 1}, for z ∈ C N (cf. [11,22,26,27]). It is a deep result that log K = V K , where…”

Section: R Pierzchała (B)mentioning

confidence: 99%

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The Łojasiewicz–Siciak Condition of the Pluricomplex Green Function

Pierzchała

2013

Potential Anal

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The aim of this paper is to address a problem raised originally by L. Gendre, later by W. Pleśniak and recently by L. Białas-Cież and M. Kosek. This problem concerns the pluricomplex Green function and consists in finding new examples of sets with so-called Łojasiewicz-Siciak ((ŁS) for short) property. So far, the known examples of such sets are rather of particular nature. We prove that each compact subset of R N , treated as a subset of C N , satisfies the Łojasiewicz-Siciak condition. We also give a sufficient geometric criterion for a semialgebraic set in R 2 , but treated as a subset of C, to satisfy this condition. This criterion applies more generally to a set in C definable in a polynomially bounded o-minimal structure.

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Section: R Pierzchała (B)mentioning

confidence: 99%

“…A spectacular example of usefulness of the extremal function is the Siciak's extension of the Bernstein-Walsh theorem to the case of several variables (cf. [26]). …”

Section: R Pierzchała (B)mentioning

confidence: 99%

The Łojasiewicz–Siciak Condition of the Pluricomplex Green Function

Pierzchała

2013

Potential Anal

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“…In particular, since the basic work of Siciak [31] and Bedford and Taylor [7], [8] a great effort was made to understand the complex MongeAmpère operator and the associated generalized Dirichlet problems (for instance, see [15], [20] and references therein).…”

Section: Introductionmentioning

confidence: 99%

The pluricomplex Poisson kernel for strongly convex domains

Bracci

Patrizio

Trapani

2008

Trans. Amer. Math. Soc.

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Let D be a bounded strongly convex domain in the complex space of dimension n. For a fixed point p epsilon partial derivative D, we consider the solution of a homogeneous complex Monge-Ampere equation with a simple pole at p. We prove that such a solution enjoys many properties of the classical Poisson kernel in the unit disc and thus deserves to be called the pluricomplex Poisson kernel of D with pole at p. In particular we discuss extremality properties (such as a generalization of the classical Phragmen-Lindelof theorem), relations with the pluricomplex Green function of D, uniqueness in terms of the associated foliation and boundary behaviors. Finally, using such a kernel we obtain explicit reproducing formulas for plurisubharmonic functions

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“…In the proof of Theorem 3 we will use the following characterization of unisolvent sets (see [12]). The mapping z -» (z, <p(fi(z)),..., 4>(fm(z))) is holomorphic, hence v is the supremum of a family of plurisubharmonic functions.…”

Section: Introductionmentioning

confidence: 99%

Joint Spectra and Analytic Set-Valued Functions

Klimek¹

1986

Transactions of the American Mathematical Society

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ABSTRACT.We investigate analyticity of joint spectra of Am-valued holomorphic mappings, where A denotes a complex Banach algebra. We show also that if K is an analytic set-valued function whose values are compact subsets of Cn and d is the transfinite diameter in Cn, then the upper-semicontinuous regularization of logd(X) is plurisubharmonic. Moreover, we give higher dimensional extensions of Aupetit's Scarcity Theorem.

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On some extremal functions and their applications in the theory of analytic functions of several complex variables

Cited by 144 publications

References 14 publications

The Łojasiewicz–Siciak Condition of the Pluricomplex Green Function

The Łojasiewicz–Siciak Condition of the Pluricomplex Green Function

The pluricomplex Poisson kernel for strongly convex domains

Joint Spectra and Analytic Set-Valued Functions

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