Alpay, Daniel; Putinar, Mihai; Vinnikov, Victor A Hilbert space approach to bounded analytic extension in the ball. (English) Zbl 1045.32015 Commun. Pure Appl. Anal. 2, No. 2, 139-145 (2003). We say that a \(C^1\)-map \(\varphi:\overline D\to\overline B\) of the closed unit disk into a unit ball in \(\mathbb{C}^n\) is an analytic disk transversally attached to the unit sphere if \(\varphi\) is holomorphic on \(D\), injective on \(\overline D\), \(\|\varphi(u)\|= 1\Leftrightarrow| u|= 1\) and \(\langle\varphi'(u),\varphi(u)\rangle\neq 0\) for \(| u|= 1\). The main result of the paper is the following: Theorem 2.3. Let \(A\) be an analytic disk in the unit ball of \(\mathbb{C}^n\), transversally attached to the unit sphere. Then any bounded analytic function on \(A\) admits an analytic extension to the Schur class of the ball. Reviewer: Sergey M. Ivashkovich (Villeneuve d’Ascq) Cited in 1 ReviewCited in 8 Documents MSC: 32D15 Continuation of analytic objects in several complex variables 32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables 32A70 Functional analysis techniques applied to functions of several complex variables 47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc. Keywords:Schur class; reproducing kernel; bounded analytic extension PDFBibTeX XMLCite \textit{D. Alpay} et al., Commun. Pure Appl. Anal. 2, No. 2, 139--145 (2003; Zbl 1045.32015) Full Text: DOI