Holomorphic convexity and Carleman approximation by entire functions on Stein manifolds

Abstract

We give necessary and sufficient conditions for totally real sets in Stein manifolds to admit Carleman approximation of class C k , k ≥ 1, by entire functions.

Carleman approximation of maps into Oka manifolds

“…Observe that the conditions of Theorem 1 are optimal due to Manne, Øvrelid and Wold's characterisation of totally real sets that admit C k -Carleman approximation [13]. By modifying our proof slightly, in a way that will be made precise in §4, we also obtain the following result.…”

“…Let M be a closed, not necessarily compact subset of X. Following [13] we make the following definitions. Define the hull of M by…”

“…The submanifold M has a Stein neighbourhood basis by [13,Corollary 3.2]. So we can assume that U is Stein and hence our theorem applies.…”

“…There are a number of generalisations of this theorem in the literature already, see Alexander [1], Gauthier and Zeron [7], Hoischen [9], Scheinberg [17], Manne [12] and Manne, Øvrelid and Wold [13]. These results concern Carleman approximation of functions.…”

“…The proof of Theorem 1, given in Section 4, involves an induction and reduction procedure making use of Poletsky's theorem [14, Theorem 3.1] as well as methods from [5] and [13]. Section 3 is dedicated to the case where the target is C; there we collect some results that will be needed in the proof of the general case of Theorem 1.…”