On functions uniquely determined by their asymptotic expansion (Q883701)

Let \(S(\alpha,\beta)\) stand for the sector \(S(\alpha,\beta)=\{z\in \mathbb{C}/0<|z|<\infty\), \(\alpha<\arg z<\beta\}\) (the number \(\beta-\alpha\) is also called the opening of the sector). Let \(\{p_0,p_1,\dots, \}\) be a sequence of complex numbers and let \(P(z)\) be a function satisfying the following conditions. 1. \(P(z)\) is analytic and single-valued in the sector \(S(\alpha,\beta)\). 2. If \({\mathcal R}_{n,P}(z)=P(z) -\sum^{n-1}_{k=0}\frac{p_k} {z^{k+1}}\) then \(\sum^\infty_{j=0} \frac{p_j}{z^{j+1}}\) is called the Gevrey asymptotic expansion of order \(k\), \(k>0\) for \(P(z)\in S(\alpha,\beta)\) if for each proper subsector \(S'\) of \(S(\alpha,\beta)\), \(\overline{S'}\subset S(\alpha, \beta\), there exist positive constants \(M(S')\) and \(a(S')\) such that \(|{\mathcal R}_{n,P}(z)|\leq M(S')\frac{(n!)^{1/k}}{(ka(S'))^n|z|^{n+1}},z\in S'\). In this paper the following theorem is proved. Theorem: Let \(P(z)\) admit a Gevrey expansion of order 1 in \(S(-\pi/2,\pi/2)\). If \(M(\delta)\) and \(a(\delta)\) satisfy \(\int_0^{\pi/2}\log\log M(\delta) \,d\delta<\infty\) and \(\frac {\delta}{a (\delta)}\to 0\) as \(\delta\to 0\) then \(P(z)\) is uniquely determined by the coefficients of its Gevrey expansion. The above theorem is based on a generalization of the following known lemma. Lemma: Suppose that \(P(z)\) is analytic in a sector \(S(\alpha,\beta)\) with opening greater than \(\pi\) and if \(|P(z)|\leq 4M\sqrt{2\pi}ae^{-a|z|}\), for \(|z|>\frac 1a\) then \(P(z)\equiv 0\).