A distributional theory for asymptotic expansions

Abstract

We present various techniques for the asymptotic expansions of generalized functions. We show that the moment asymptotic expansions hold for a very wide variety of kernels such as generalized functions of rapid decay and rapid oscillations. We do not use Mellin transform techniques as done by previous authors in the field. Instead, we introduce a direct approach that not only solves the one-dimensional problems but also applies to various multidimensional integrals and oscillatory kernels as well. This approach also helps in the development of various asymptotic series arising in diverse fields of mathematics and physics. We find that the asymptotic expansions of generalized functions depend on the selection of suitable spaces of test functions. Accordingly, we have exercised special care in classifying the spaces and the distributions defined on them. Furthermore, we use the theory of topological tensor products to obtain the expansions of vector-valued distributions. We present several examples to illustrate that many classical results follow in a simple manner. For instance, we derive from our results the asymptotic expansions of certain series considered by Ramanujan.