We provide Abelian and Tauberian theorems for wavelet and regular-
izing transforms of Banach-valued tempered distributions, that is, trans-
forms of the form $M_{phi}^f (x, y) = (f ∗ phi_y )(x),$ where $phi$ is a
test function and $phi_y (·) = y^{-n} phi(·/y). If the first moment of $phi$
vanishes it is a wavelet type ransform; otherwise, we say it is a non-wavelet
type (regularizing) transform. Both cases at finite points and at infinity are studied.
It is shown that the asymptotic properties of distributions can be completely
characterized by boundary asymptotics of the wavelet and non-wavelet
transforms plus natural Tauberian hypotheses. We apply Tauberian type
results for Banach-valued distributions to spaces introduced by Mayer
and Bony in the local and microlocal analysis, to regularity theory within
generalized function algebras, to the pointwise analysis of the family of
Riemann's functions and distributions at the rational points, and to Tauberian
theorems for Laplace transforms of distributions with supports in cones,
for example, a new proof of the Littlewood Tauberian theorem.
Joint results of J. Vindas and S. Pilipović.
