Gérard-Sibuya's versus Majima's concept of asymptotic expansion in several variables (Q2748388)

For \(j=1,\dots, n\), an open sector in \(\mathbb{C}\) with the vertex at the origin is given by NEWLINE\[NEWLINE S_j=\left\{z\in\mathbb{C}: \theta_{1j}<\arg z<\theta_{2j},\;0<|z|<R_j\right\}, NEWLINE\]NEWLINE with \(0<\theta_{2j}-\theta_{1j}\leq 2\pi\) and \(R_j\in (0,+\infty]\). The Cartesian product \(S=\prod_{j=1}^n S_j\subset\mathbb{C}^n\) is called (open) polysector in \(\mathbb{C}^n\) with vertex at the origin. A polysector \(T=\prod_{j=1}^n T_j\) in \(\mathbb{C}^n\) (with vertex at the origin) is a bounded proper subpolysector of \(S\) if it is bounded and \(\overline{T_j}\subset S_j\cup\{0\}\), \(j=1,\dots,n\). In this case, we write \(T\ll S\). NEWLINENEWLINENEWLINEA holomorphic function \(f: S\to\mathbb{C}\) admits a Gérard-Sibuya asymptotic expansion at the origin if there exists a (formal) power series \(\sum_{\alpha\in\mathbb{Z}_+^n}a_\alpha z^\alpha\), with \(a_\alpha\in\mathbb{C}\), such that for every \(T\ll S\) and \(m\in\mathbb{Z}_+\) there exists a constant \(C>0\) (depending on \(T\) and \(m\)) such that NEWLINE\[NEWLINE\Biggl|f(z)-\sum_{j=0}^m \sum_{|\alpha|=j} a_\alpha z^\alpha \Biggr|\leq C\|z\|^{m+1},\quad z\in T. NEWLINE\]NEWLINE Here, \(\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb{Z}_+^n\), \(|\alpha|=\sum_{j=1}^n\alpha_j\), \(z=(z_1,\dots,z_n)\in\mathbb{C}^n\), \(z^\alpha=z_1^{\alpha_1}\cdot\dots\cdot z_n^{\alpha_n}\). NEWLINENEWLINENEWLINEIn the one-dimensional case, the following two properties hold true [see \textit{W. Wasow}, Asymptotic expansions for ordinary differential equations. New York-London-Sydney: Interscience Publishers, a division of John Wiley \& Sons (1965; Zbl 0133.35301)]: NEWLINENEWLINENEWLINE(A) If a function \(f\) admits an asymptotic expansion then its derivatives admit such expansion. NEWLINENEWLINENEWLINE(B) \(f\) admits an asymptotic expansion if and only if its derivatives (including \(f=f^{(0)}\)) are bounded on bounded proper subsectors of \(S\). NEWLINENEWLINENEWLINEThe article is devoted to the construction of an example of a holomorphic function, admitting a Gérard-Sibuya asymptotic expansion on a polysector of \(\mathbb{C}^n\), \(n\geq 2\), which satisfies neither property (A) nor (B). NEWLINENEWLINENEWLINEThe authors provide a proof of the equivalence of the following statements: NEWLINENEWLINENEWLINE(i) the derivatives of \(f:S\subset\mathbb{C}^n\to\mathbb{C}\) admit an Gérard-Sibuya asymptotic expansion at the origin; NEWLINENEWLINENEWLINE(ii) the derivatives of \(f\) are bounded on bounded proper subpolysectors of \(S\); NEWLINENEWLINENEWLINE(iii) \(f\) is strongly asymptotically expansible, that is, in the sense of Majima. NEWLINENEWLINENEWLINEThus, Majima's definition is more restrictive than Gérard-Sibuya's one, and it implies the properties (A) and (B). NEWLINENEWLINENEWLINELet us note that the authors use the definition of strongly asymptotically expansible function which is an adaptation of the concept of strongly asymptotic expansion [see \textit{H. Majima}, Funkc. Ekvacioj, Ser. Int. 26, 131-154 (1983; Zbl 0533.32001)] to the context of the present article. NEWLINENEWLINENEWLINEThe main results are formulated for the spaces \(\mathcal{A}(S,E)\) of functions defined and holomorphic on \(S\) with values in a Fréchet space \(E\), and those derivatives remain bounded on bounded proper subpolysectors of \(S\). Some differential properties of functions from these spaces are also investigated.