Interpolation by holomorphic functions smooth to the boundary in the unit ball of \({\mathbb{C}}^ n\)
type: Item
instance of: scholarly article
title: Interpolation by holomorphic functions smooth to the boundary in the unit ball of \({\mathbb{C}}^ n\)
zbMATH Open document ID: 0585.32018
DOI: 10.1007/BF01458591
author: Joaquim Bruna
author: J. M. Ortega
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164116
review text: [Shortened version of the original review]. Let B denote the unit ball in \({\mathbb{C}}^ n\) and S its boundary; let \(A^{m,s}=H(D)\cap C^{m,s}(\bar D)\) and \(A^{\infty}=\cap_{m,s}A^{m,s}\). The paper concerns with interpolation by functions in \(A^{m,s}\) from closed sets \(E\subset S.\) For \(n>1\) and \(m=\infty\) there were given partial descriptions of this sets by Hakim, Sibony, Chaumat and Chollet. The aim of this work is to treat the intermediate case \(0<m+s<+\infty\). In this context it is natural to define, similarly as Whitney's definition in the real case, the space \(A^{m,s}(E)\) of non-isotropic holomorphic jets of class \(A^{m,s}\) on a closed set \(E\subset S\) in order to define E to be an \(A^{m,s}\)-interpolation set. With the definition of a K-set, given by \textit{J. Chaumat} and \textit{A. M. Chollet} in ''Ensembles de zéros et d'interpolation à la frontière de domaines strictement pseudoconvex'', Preprint (1984) the main results are: Theorem: A closed set E contained in a transverse curve \(\Gamma\) is an interpolation set for all the algebras \(A^{m,s}\), \(0<m+s<\infty\) if and only if it is a K-set. Theorem: Any closed set E contained in a complex-tangential submanifold of S is an \(A^{m,s}\)-interpolation set \(0<m+s<\infty.\) For \(m=\infty\) the authors also study the division property where E has the division property for \(A^{\infty}\) if, given arbitrary \(f_ 1,...,f_ r\) functions in \(C^{\infty}(\bar B)\) flat on E, there is a \(g\in A^{\infty}(\bar D)\) flat on E such that \(f_ i=h_ i\cdot g\). The main results for \(A^{\infty}\) are then: Every set with the division property for \(A^{\infty}\) is an \(A^{\infty}\)-interpolation set and every K-set has the division property.
Mathematics Subject Classification ID: 32A38
Mathematics Subject Classification ID: 32D15
Mathematics Subject Classification ID: 32A40
zbMATH DE Number: 3937533
zbMATH Keywords: algebra of holomorphic functions in the unit ball
zbMATH Keywords: non-isotropic holomorphic jets
zbMATH Keywords: interpolation set
zbMATH Keywords: K-set
zbMATH Keywords: division property
MaRDI profile type: Publication
cites work: The interpolating sets for \(A^ \infty\)
cites work: Boundary Interpolation Sets for Holomorphic Functions Smooth to the Boundary and BMO
cites work: Closed finitely generated ideals in algebras of holomorphic functions and smooth to the boundary in strictly pseudoconvex domains
cites work: Formules explicites pour les solutions minimales de l'équation \(\overline\partial u=f\) dans la boule et dans le polydisque de \(\mathbb C^n\)
cites work: Ensembles pics pour \(A^\infty(D)\)
cites work: Caractérisation et propriétés des ensembles localement pics de \(A^\infty(D)\)
cites work: Ensembles de zéros et d'interpolation à la frontière de domaines strictement pseudoconvexes. (Zero sets and interpolation on the boundary of strictly pseudoconvex domains)
cites work: Q2548398
cites work: Q3922907
cites work: Ensembles pics dans des domaines strictement pseudo-convexes
cites work: Holomorphic approximation and hyperfunction theory on a C\(^1\) totally real submanifold of a complex manifold
cites work: Boundary properties of holomorphic functions of several complex variables
cites work: Q5566770
cites work: Cauchy transforms of measures, and a characterization of smooth peak interpolation sets for the ball algebra
cites work: Q3958740
cites work: Singular integrals and estimates for the Cauchy-Riemann equations
OpenAlex ID: W1999930009
description: scientific article; zbMATH DE number 3937533