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.