Let X be a nonsingular complex algebraic curve and let Y be a nonsingular rational complex algebraic surface. Given a compact subset K of X, every holomorphic map from a neighborhood of K in X into Y can be approximated by rational maps from X into Y having no poles in K. If Y is a nonsingular projective complex surface with the first Betti number nonzero, then such an approximation is impossible.

Let X be an analytic set defined by polynomials whose coefficients $a₁,...,a_s$ are holomorphic functions. We formulate conditions on sequences $a_{1,\nu },...,a_{s,\nu }$ of holomorphic functions converging locally uniformly to $a₁,...,a_s$, respectively, such that the sequence $X_{\nu }$ of sets obtained by replacing $a_j$’s by $a_{j,\nu }$’s in the polynomials converges to X.

Boundary values of zero-smooth Besov analytic functions in the unit ball of ${ℂ}^n$ are investigated. Bounded Besov functions with prescribed lower semicontinuous modulus are constructed. Correction theorems for continuous Besov functions are proved. An approximation problem on great circles is studied.

Let Ω be a bounded pseudo-convex domain in Cn with a C∞ boundary, and let S be the set of strictly pseudo-convex points of ∂Ω. In this paper, we study the asymptotic behaviour of holomorphic functions along normals arising from points of S. We extend results obtained by M. Ortel and W. Schneider in the unit disc and those of A. Iordan and Y. Dupain in the unit ball of Cn. We establish the existence of holomorphic functions of given growth having a "prescribed behaviour" in almost all normals arising...

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...

We first give a general growth version of the theorem of Bernstein-Walsh-Siciak concerning the rate of convergence of the best polynomial approximation of holomorphic functions on a polynomially convex compact subset of an affine algebraic manifold. This can be considered as a quantitative version of the well known approximation theorem of Oka-Weil. Then we give two applications of this theorem. The first one is a generalization to several variables of Winiarski's theorem relating the growth of...

Soit $\Omega$ un domaine d’holomorphie de ${\mathbf{C}}^n$ et soit $\psi$ une fonction positive telle que pour tout $k\>0$ on ait$$\underset{z\in \Omega }{sup}\left\{\psi \right(z),[\min \mathrm{dist}(z,{\mathbf{C}}^n\setminus \Omega ),(1+|z{|}^2{)}^{-1/2}{]}^k\}\<\infty .$$On note $H^p(\Omega ,\psi )$, $1\le p\le \infty$, l’espace des fonctions $f$ holomorphes dans $\Omega$ telles que $\parallel f{\parallel }_p={\left({\int }_{\Omega }|f{|}^p\psi \right)}^{1/p}\<\infty$. 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...

Soit $\Sigma$ une sous-variété de ${\mathbf{C}}^n$, de classe $C^{\infty }$ et totalement réelle. Si $w$ est une fonction continue strictement positive sur $\Sigma$, on désigne par $C_w\left(\Sigma \right)$ l’espace des fonctions $f$ continues sur $\Sigma$ telles que $w\left|f\right|$ tend vers zéro à l’infini. On munit cet espace de la norme $\parallel f\parallel ={sup}_{x\in \Sigma }w\left(x\right)\left|f\left(x\right)\right|$ et on suppose qu’il contient les polynômes. Sous des hypothèses de nature géométrique sur $\Sigma$, on donne des conditions suffisantes pour l’approximation des fonctions de $C_w\left(\Sigma \right)$ par des fonctions holomorphes au voisinage de $\Sigma$ ou par des polynômes.