Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multidimensional Tauberian theorems for wavelet and non-wavelet
  transforms   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.

1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph