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Weierstrass theorems in strong asymptotic analysis. (English) Zbl 0988.54019

The study of strong asymptotic developments over open polysectors \(V\) in complex \(n\)-space \(\mathbb{C}^n\) was given a new direction by 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 \(V\).

MSC:

54C40 Algebraic properties of function spaces in general topology
46E25 Rings and algebras of continuous, differentiable or analytic functions
32A30 Other generalizations of function theory of one complex variable
30E15 Asymptotic representations in the complex plane

Citations:

Zbl 0533.32001
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