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Our goal in this article is to derive Abelian theorems for the two-sided Laplace transform, Mellin transform, one-sided real Laplace transform and Stieltjes transform over distributions of compact support and over certain function spaces of generalized functions. [ABSTRACT FROM AUTHOR]

We make a complete wavelet analysis of asymptotic properties of distributions. The study is carried out via Abelian and Tauberian type results, connecting the boundary asymptotic behavior of the wavelet transform with local and non-local quasiasymptotic properties of elements in the Schwartz class of tempered distributions. Our Tauberian theorems are full characterizations of such asymptotic properties. We also provide precise wavelet characterizations of the asymptotic behavior of elements in the dual of the space of highly time-frequency localized functions over the real line. For the use of the wavelet transform in local analysis, we study the problem of extensions of distributions initially defined on ℝ∖{0} to ℝ; in this extension problem, we explore the asymptotic properties of extensions of a distribution having a prescribed asymptotic behavior. Our results imply intrinsic properties of functions and measures as well, for example, we give a new proof of the classical Littlewood Tauberian theorem for power series. [ABSTRACT FROM AUTHOR]

We establish some assertions of Tauberian and Abelian types which enable us to find connections between the asymptotic properties of the Laplace transform at infinity and the asymptotics of the corresponding densities of rapidly decaying distributions (at infinity or in some neighborhood of zero). As applications of our Tauberian type theorems we present asymptotics for the density f ( α,ρ)( x) of “extreme” stable laws with parameters ( α, ρ) for ρ = ±1 and x lying in the domain of rapid decay of f ( α,ρ)( x). This asymptotics had been found in [1–5] by a more complicated method. [ABSTRACT FROM AUTHOR]