Abstract
Suppose f is a C∞ function or a power series in n variable belonging to a Carleman or a Beurling class Fn. Let Γ be a family of polynomial maps P:Cd → Cn, d ≤ n, such that f ∘ P∈Fd, ∀ P∈Γ. Under what conditions can one conclude that f∈Fn? The Fn-analogs of the following theorems are obtained: (i) The Bochnak–Siciak theorem: A C∞ function on Rn that is analytic on every line is analytic. (ii) Zorn´s theorem: If a double power series F(x,y) has the property that for all ξ,η∈C the t-series F(ξt,ηt) converges, then F(x,y) is convergent as a double series. The methods applied here also yield new proofs of the above mentioned theorems as well as their improvements due to P. Lelong ([9]), Abhyankar, Moh, and Sathye ([1,19]), and Levenberg and Molzon ([10]).
© by Oldenbourg Wissenschaftsverlag, München, Germany
