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A method of holomorphic retractions and pseudoinverse matrices in the theory of continuation of \(\delta\)-tempered functions. (English) Zbl 0639.32004

It is given a thorough investigation of some problems from theory of continuation of holomorphic functions with restricted growth. Let \(\delta: X\to (0,+\infty)\) be a bounded function. If \({\mathcal O}^{(k)}(X,\delta)\) is the vector space of all functions f holomorphic on the complex analytic space X, countable at infinity, for which \(\delta^ kf\) is bounded, then \({\mathcal O}(X,\delta)=\cup_{k\geq 0}{\mathcal O}^{(k)}(X,\delta)\) is the space of functions with restricted growth. If X is a Stein domain, \(\delta\) is a regular weight function and M is an analytic submanifold of X, determined by functions from \({\mathcal O}(X,\delta)\), then it is proved that each bounded holomorphic extension operator \(T: {\mathcal O}(M,\delta)\to {\mathcal O}(X,\delta)\) with \(R_ m\circ T=id\), \(R_ M\) being the restriction operator, is given as: \(Tf=f\circ \pi\), where \(\pi\) : \(X\to M\) is a suitably chosen holomorphic retraction. Conditions for existence of linear continuous extension operators that control the functions’ growth are found.
Reviewer: T.Tonev

MSC:

32A40 Boundary behavior of holomorphic functions of several complex variables
32A10 Holomorphic functions of several complex variables