Existence and uniqueness of v-asymptotic expansions and Colombeau's generalized numbers

[摘要] We define a type of generalized asymptotic series called v-asymptotic. We show that every function with moderate growth at infinity has a v-asymptotic expansion. We also describe the set of v-asymptotic series, where a given function with moderate growth has a unique v-asymptotic expansion. As an application to random matrix theory we calculate the coefficients and establish the uniqueness of the v-asymptotic expansion of an integral with a large parameter. As another application (with significance in the non-linear theory of generalized functions) we show that every Colombeau's generalized number has a v-asymptotic expansion. A similar result follows for Colombeau's generalized functions, in particular, for all Schwartz distributions. (c) 2005 Elsevier Inc. All rights reserved.

[发布日期] 2005-12-01

[关键词] asymptotic expansion;valuation;ultrametric space;colombeau generalized functions;random matrix theory