Holomorphic approximation in Fréchet spaces (Q2386761)

This paper is concerned with the uniform approximation of holomorphic functions defined on an open subset of a class of Fréchet spaces by entire holomorphic functions. The spaces considered are `generalised sequence Fréchet spaces'. Given a set \(\Gamma\) and a function \(p: {\mathbb R}\times \Gamma\to {\mathbb C}\) with \(\log p(\cdot,\gamma)\) even and convex for every \(\gamma\) in \(\Gamma\), the author introduces the space \(X\) of all \(x: \Gamma\to {\mathbb C}\) such that \(\| x\| _\theta:=\sum_\gamma p(\theta,\gamma)| x(\gamma)| <\infty\) for all \(\theta\), endowed with the topology generated by the \(\| \cdot\| _\theta\). This class includes \(\ell_1\), the space of rapidly decreasing sequences, \(s\), and the Fréchet nuclear space \({\mathcal H}({\mathbb C})\) endowed with the compact open topology. The author proves that given any space \(X\) in this class, given \(0<r<R\) and any real \(\theta\), that any holomorphic function on \(\{x\in X:\| x\| _\theta<R\}\) can be uniformly approximated by entire holomorphic functions on \(\{x\in X:\| x\| _\theta<r\}\). This result is actually obtained in the more general but also more technical setting where \(\Gamma\) has the structure of a finite-dimensional Stein manifold and is obtained using a delicate examination of the monomial expansion of holomorphic functions. The results of this paper are an extension of the special case \(X=\ell_1\) obtained by the author in [\textit{L. Lempert}, Ann. Inst. Fourier Grenoble 49, 1293--1304 (1999; Zbl 0944.46046)]. As a converse, the author proves that, if \(X\) is a Fréchet space such that every neighbourhood \(U\) of \(0\) admits a neighbourhood \(W\) of \(0\) with the property that each holomorphic function on \(U\) can be uniformly approximated by entire holomorphic functions on \(W\), then \(X\) is a DN space in the sense of \textit{D. Vogt} [Manuscr. Math. 17, 267--290 (1975; Zbl 0349.46040)].