Weierstrass theorems in strong asymptotic analysis (Q2770176)

The study of strong asymptotic developments over open polysectors \(V\) in complex \(n\)-space \(\mathbb{C}^n\) was given a new direction by \textit{H. Majima} [Funkc. Ekvacioj, Ser. Int. 26, 131-154 (1983; Zbl 0533.32001)]. If \(f\) is a holomorphic function on \(V\), then a polynomial approximant to \(f\) is defined on \(V\) in such a way that \(f\) is said to admit a strong asymptotic development. We denote the set of such \(f\) by \(A(V)\). We say that \(W\) is a sub-polysector of \(V\) and write \(W<V\) in case the closure in \(V\) of \(W\) is a closed polysector. In the present paper, the author gives a new proof of the result that \(f\in A(V)\) if, and only if, for any \(W<V\), \(f|_W\) admits a \(C^\infty\) real-valued extension to \(\mathbb{C}^n\). He proves an implicit function theorem and Weierstrass-type asymptotic preparation and division theorems in the setting of \(A(V)\) with some restrictions on the dimension and shape of \(V\).