We apply pluripotential theory to establish results in ${ℝ}^k$ concerning uniform approximation by functions of the form wⁿPₙ where w denotes a continuous nonnegative function and Pₙ is a polynomial of degree at most n. Then we use our work to show that on the intersection of compact sections $\Sigma \subset {ℝ}^k$ a continuous function on Σ is uniformly approximable by θ-incomplete polynomials (for a fixed θ, 0 < θ < 1) iff f vanishes on θ²Σ. The class of sets Σ expressible as the intersection of compact sections includes...

In this paper we obtain results on approximation, in the multidimensional complex case, of functions from ${}^{\infty }\left(K\right)$ by complex polynomials. In particular, we generalize the results of Pawłucki and Pleśniak (1986) for the real case and of Siciak (1993) in the case of one complex variable. Furthermore, we extend the results of Baouendi and Goulaouic (1971) who obtained the order of approximation in the case of Gevrey classes over real compacts with smooth analytic boundary and we present the orders of approximation...

Soit $\Omega$ un domaine d’holomorphie de ${\mathbf{C}}^n$ et soit $\psi$ une fonction positive telle que pour tout $k\&gt;0$ on ait$$\underset{z\in \Omega }{sup}\left\{\psi \right(z),[\min \mathrm{dist}(z,{\mathbf{C}}^n\setminus \Omega ),(1+|z{|}^2{)}^{-1/2}{]}^k\}\&lt;\infty .$$On note $H^p(\Omega ,\psi )$, $1\le p\le \infty$, l’espace des fonctions $f$ holomorphes dans $\Omega$ telles que $\parallel f{\parallel }_p={\left({\int }_{\Omega }|f{|}^p\psi \right)}^{1/p}\&lt;\infty$. On donne des conditions nécessaires et des conditions suffisantes pour l’approximation des fonctions de $H^p(\Omega ,\psi )$ par des fonctions holomorphes dans un ouvert contenant $\Omega$, ou par des polynômes. On obtient comme cas particulier les résultats suivants :a) les polynômes sont denses dans $H^p(\Omega ,\exp (-\Phi \left)\right)$ lorsque $\Omega$ est ouvert convexe (non borné) et $\Phi$ une fonction...

We study the singularities of plurisubharmonic functions using methods from convexity theory. Analyticity theorems for a refined Lelong number are proved.

We introduce the notion of the Shilov boundary for some subfamilies of upper semicontinuous functions on a compact Hausdorff space. It is by definition the smallest closed subset of the given space on which all functions of that subclass attain their maximum. For certain subfamilies with simple structure we show the existence and uniqueness of the Shilov boundary. We provide its relation to the set of peak points and establish Bishop-type theorems. As an application we obtain a generalization of...

We show that in the class of compact, piecewise $C^1$ curves K in ${ℝ}^n$, the semialgebraic curves are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for the derivatives of (the traces of) polynomials on K.

We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity, with a coefficient involving the Kählerian volume of the real locus of the manifold as well as the expected determinant of random real symmetric matrices of given index. In particular, for large dimensions, these coefficients get exponentially small away from...

We survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ 0,1. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves...

We study boundary values of functions in Cegrell’s class ${}_{\psi }$.

We describe compact subsets K of ∂𝔻 and ℝ admitting holomorphic functions f with the domains of existence equal to ℂ∖K and such that the pluripolar hulls of their graphs are infinitely sheeted. The paper is motivated by a recent paper of Poletsky and Wiegerinck.

We study the B-regularity of some classes of domains in ℂⁿ. The results include a complete characterization of B-regularity in the class of Reinhardt domains, we also give some sufficient conditions for Hartogs domains to be B-regular. The last result yields sufficient conditions for preservation of B-regularity under holomorphic mappings.

Given a holomorphic mapping $f:{\mathbb{P}}^2\to {\mathbb{P}}^2$ of degree $d\ge 2$ we give sufficient conditions on a positive closed (1,1) current of $S$ of unit mass under which $d^{-n}{f}^{n*}S$ converges to the Green current as $n\to \infty$. We also conjecture necessary condition for the same convergence.

We apply the Cauchy-Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set E in ${ℝ}^N$ is Hölder continuous then E admits a Szegö type inequality with weight function $dist{(x,\partial E)}^{-(1-\kappa )}$ with a positive κ. This can be viewed as a (nontrivial) generalization of the classical result for the interval E = [-1,1] ⊂ ℝ.

We study Cegrell classes on compact Kähler manifolds. Our results generalize some theorems of Guedj and Zeriahi (from the setting of surfaces to arbitrary manifolds) and answer some open questions posed by them.