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

Details Existence and uniqueness of $v$-asymptotic expantions and Colombeau's generalized numbers Existence and uniqueness of $v$-asymptotic expantions and Colombeau's generalized numbers

2006-01-30

arxiv.org/abs/math/0601720v1

Journal of Mathematical Analysis and Applications, Vol. 312, Issue 1, 2005, p. 261-279.

Mathematics

Classical Analysis and ODEs

19 pages

Scientific paper

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.

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