Multidimensional Tauberian theorems for wavelet and non-wavelet transforms

Abstract

We study several Tauberian properties of regularizing transforms of tempered distributions 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 the zeroth moment of $\phi$ vanishes, it is a wavelet type transform; otherwise, we say it is a non-wavelet type transform. The first aim of this work is to show that the scaling (weak) asymptotic properties of distributions are \emph{completely} determined by boundary asymptotics of the regularizing transform plus natural Tauberian hypotheses. Our second goal is to characterize the spaces of Banach space-valued tempered distributions in terms of the transform $M^{\mathbf{f}}_{\phi}(x,y)$. We investigate conditions which ensure that a distribution that a priori takes values in locally convex space actually takes values in a narrower Banach space. 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, and develop a number of techniques which are specially useful when applied to scalar-valued functions and distributions. Among such applications, we obtain the full weak-asymptotic series expansion of the family of Riemann-type distributions $R_{\beta}(x)=\sum_{n=1}^{\infty}e^{i\pi xn^{2}}/n^{2\beta}$, $\beta\in\mathbb{C}$, at every rational point. We also apply the results to regularity theory within generalized function algebras, to the stabilization of solutions for a class of Cauchy problems, and to Tauberian theorems for the Laplace transform.