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We study several Tauberian properties of regularizing transforms of tempereddistributions with values in Banach spaces, that is, transforms of the form$M^{\mathbf{f}}_{\phi}(x,y)=(\mathbf{f}\ast\phi_{y})(x)$, where the kernel$\phi$ is a test function and $\phi_{y}(\cdot)=y^{-n}\phi(\cdot/y)$. If thezeroth moment of $\phi$ vanishes, it is a wavelet type transform; otherwise, wesay it is a non-wavelet type transform.  The first aim of this work is to show that the scaling (weak) asymptoticproperties of distributions are \emph{completely} determined by boundaryasymptotics of the regularizing transform plus natural Tauberian hypotheses.Our second goal is to characterize the spaces of Banach space-valued tempereddistributions in terms of the transform $M^{\mathbf{f}}_{\phi}(x,y)$. Weinvestigate conditions which ensure that a distribution that a priori takesvalues in locally convex space actually takes values in a narrower Banachspace. Special attention is paid to find the \emph{optimal} class of kernels$\phi$ for which these Tauberian results hold.  We give various applications of our Tauberian theory in the pointwise and(micro-)local regularity analysis of Banach space-valued distributions, anddevelop a number of techniques which are specially useful when applied toscalar-valued functions and distributions. Among such applications, we obtainthe full weak-asymptotic series expansion of the family of Riemann-typedistributions $R_{\beta}(x)=\sum_{n=1}^{\infty}e^{i\pi xn^{2}}/n^{2\beta}$,$\beta\in\mathbb{C}$, at every rational point. We also apply the results toregularity theory within generalized function algebras, to the stabilization ofsolutions for a class of Cauchy problems, and to Tauberian theorems for theLaplace transform.

Multidimensional Tauberian theorems for wavelet and non-wavelet transforms