Elementary acceleration and multisummability. I
Elementary acceleration and multisummability. I
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基本加速和多重可求性。
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发表时间：
1990
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通讯作者：
J. Ramis
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作者：
Jean Martinet;J. Ramis
本文摘自同一作者即将出版的一本书的内容[MR 3]。与[MR 2]第2章相连的第1段至第3段或多或少形成了一个独立的集合。我们回顾了Borel可和性的基本定义（Borel [Bo 1]，[Bo 2]），以及它的自然推广k-可和性（Leroy [Le]，Nevanlinna [Ne]，Ramis [Ra 1]）。我们描述了Ecalle [E4]引入的初等加速度以及与之相关的不同的可和性算子。如果与[E4]相比，我们的描述略有修改，以符合我们的几何解释[MR 2]，[MR 3]。在第4段中，作为应用的一个例子，我们使用Ramis [Ra 3]的一个结果给出了Stokes乘数的一个自然的、简单的和一般的定义。也是[Ra 2]），并给出了Ramis（[Ra 4]，[Ra 5]）关于线性微分方程微分Galois群计算的一个定理的新证明.作为副产品，我们也得到了亚纯分类的亚纯线性微分方程的黎曼曲面的有限维线性表示的野生基本群。(This是黎曼-希尔伯特对应的自然推广。）第6段是非常粗略的;我们描述解析几何的无穷小邻域（遵循Deligne [De 4]的思想）和解析函数（弱解析函数和野生解析函数）在这些邻域上的层。之后，我们能够给出几何解释的基本加速度，求和和斯托克斯现象，并得到各种推广（一个正式的幂级数的总和现在是一个野生解析函数）重要的扩展到非线性的情况
This paper is extracted from the contents of a forthcoming book by the same authors [MR 3]. Paragraphs 1 to 3 joined to chapter 2 of [MR 2] form a more or less self-contained set. We recall basic definitions about Borel-summability (Borel [Bo 1], [Bo 2]), and its natural generalization k-summability (Leroy [Le], Nevanlinna [Ne], Ramis [Ra 1]). We describe the elementary acceleration introduced by Ecalle [E 4] and different summability operators related to it. If one compares to [E 4] our description is slightly modified in order to fit with our geometric interpretations [MR 2], [MR 3]. In paragraph 4 as an example of application we give a natural, simple and general, definition of Stokes multipliers, using a result of Ramis [Ra 3] (cf. also [Ra 2]), and derive a new proof of a theorem of Ramis ([Ra 4], [Ra 5]) about the computation of the differential galois group of a linear differential equation. As a byproduct we get also a description of the meromorphic classification of meromorphic linear differential equations on a Riemann surface by the finite dimensional linear representations of a wild fundamental group. (This is a natural generalization of the Riemann-Hilbert correspondence.) Paragraph 6 is very sketchy; we describe infinitesimal neighbourhoods of analytic geometry (following an idea of Deligne [De 4]) and sheaves of analytic functions (weakly analytic and wild analytic functions) on these neighbourhoods. Afterwards we are able to give geometric interpretations of elementary acceleration, summability and Stokes phenomena and to get various generalizations (the sum of a formal power series is now a wild analytic function) important for extensions to non-linear situations