关于一个Watson-like唯一性定理和Gevrey展开式
On a Watson-like Uniqueness Theorem and Gevrey Expansions
摘要：
本文给出了一类极大的解析函数，其元素与其渐近展开一一对应。近几十年来，它已经实现了（B。Malgrange，J. Ecalle，J. P. Ramis，Y. Sibuya等人），许多普通（甚至非线性）解析微分方程的形式幂级数解实际上是正则解的Gevrey展开式。沃森的唯一性定理属于这个新理论的基础。本文讨论了在复平面的角区域上允许Gevrey展开的函数类的Watson唯一性定理的一个推广，其开度小于或等于（\frac πk），其中（k）是Gevrey展开的阶.我们提出的条件，确保唯一性，并建议扩展沃森的表示定理。这些结果可应用于某些类型的微分方程的解决方案，以获得最佳的精度估计的解决方案从相应的Gevrey展开的有限和的偏差。
英文摘要：
We present a maximal class of analytic functions, elements of which are in one-to-one correspondence with their asymptotic expansions. In recent decades it has been realized (B. Malgrange, J. Ecalle, J.-P. Ramis, Y. Sibuya et al.), that the formal power series solutions of a wide range of systems of ordinary (even non-linear) analytic differential equations are in fact the Gevrey expansions for the regular solutions. Watson's uniqueness theorem belongs to the foundations of this new theory. This paper contains a discussion of an extension of Watson's uniqueness theorem for classes of functions which admit a Gevrey expansion in angular regions of the complex plane with opening less than or equal to (\frac πk,) where (k) is the order of the Gevrey expansion. We present conditions which ensure uniqueness and which suggest an extension of Watson's representation theorem. These results may be applied for solutions of certain classes of differential equations to obtain the best accuracy estimate for the deviation of a solution from a finite sum of the corresponding Gevrey expansion.
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