Structural theorems for quasiasymptotics of distributions at the origin

Abstract

An open question concerning the quasiasymptotic behavior of distributions at the origin is solved. The question is the following: Suppose that a tempered distribution has quasiasymptotic at the origin in S′(R{double-struck}), then the tempered distribution has quasiasymptotic in D′(R{double-struck}), does the converse implication hold? The second purpose of this article is to give complete structural theorems for quasiasymptotics at the origin. For this purpose, asymptotically homogeneous functions with respect to slowly varying functions are introduced and analyzed. © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.