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Markov inequality on sets with polynomial parametrization

Mirosław Baran (1994)

Annales Polonici Mathematici

The main result of this paper is the following: if a compact subset E of n is UPC in the direction of a vector v S n - 1 then E has the Markov property in the direction of v. We present a method which permits us to generalize as well as to improve an earlier result of Pawłucki and Pleśniak [PP1].

Markov's property for kth derivative

Mirosław Baran, Beata Milówka, Paweł Ozorka (2012)

Annales Polonici Mathematici

Consider the normed space ( ( N ) , | | · | | ) of all polynomials of N complex variables, where || || a norm is such that the mapping L g : ( ( N ) , | | · | | ) f g f ( ( N ) , | | · | | ) is continuous, with g being a fixed polynomial. It is shown that the Markov type inequality | / z j P | | M ( d e g P ) m | | P | | , j = 1,...,N, P ( N ) , with positive constants M and m is equivalent to the inequality | | N / z . . . z N P | | M ' ( d e g P ) m ' | | P | | , P ( N ) , with some positive constants M’ and m’. A similar equivalence result is obtained for derivatives of a fixed order k ≥ 2, which can be more specifically formulated in the language of normed algebras. In...

Matrix inequalities and the complex Monge-Ampère operator

Jonas Wiklund (2004)

Annales Polonici Mathematici

We study two known theorems regarding Hermitian matrices: Bellman's principle and Hadamard's theorem. Then we apply them to problems for the complex Monge-Ampère operator. We use Bellman's principle and the theory for plurisubharmonic functions of finite energy to prove a version of subadditivity for the complex Monge-Ampère operator. Then we show how Hadamard's theorem can be extended to polyradial plurisubharmonic functions.

Meilleure approximation polynomiale et croissance des fonctions entières sur certaines variétés algébriques affines

Ahmed Zeriahi (1987)

Annales de l'institut Fourier

Soit K un compact polynomialement convexe de C n et V K son “potentiel logarithmique extrémal” dans C n . Supposons que K est régulier (i.e. V K continue) et soit f une fonction holomorphe sur un voisinage de K . On construit alors une suite { P } 1 de polynôme de n variables complexes avec deg ( P ) pour 1 , telle que l’erreur d’approximation max z K | f ( z ) - P ( z ) | soit contrôlée de façon assez précise en fonction du “pseudorayon de convergence” de f par rapport à K et du degré de convergence . Ce résultat est ensuite utilisé pour étendre...

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