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Differentiable interpolation on the polydisc. (English) Zbl 0586.32020

Let \(A^ p(U^ N)\) denote the algebra of functions holomorphic in the polydisc \(U^ N\) whose pth order derivatives extend continuously to the closure of \(U^ N\) and let \(A^{p,\alpha}(U^ N)\) be the subalgebra of \(A^ p(U^ N)\) consisting of those functions whose pth order derivatives satisfy a Hölder condition of order \(\alpha\) on \(U^ N\). If f is a function of class \(C^ p\) on the distinguished boundary \(T^ N\) of \(U^ N\) and if a subset E of \(T^ N\) is a peak set for the algebra \(A^ p(U^ N)\), then the authors show that there exists a function g in \(A^{p-1,\alpha}(U^ N)\) for all \(\alpha\) between zero and one such that \(g=f\) on E. The authors do not know whether this loss of smoothness is intrinsic to the problem or whether they simply stem from the methods they use.
Reviewer: P.M.Gauthier

MSC:

32A38 Algebras of holomorphic functions of several complex variables
32A40 Boundary behavior of holomorphic functions of several complex variables
32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
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