## Tauberian theorems for the wavelet transform.(English)Zbl 1219.42030

The goal of this paper is to study, via Abelian-Tauberian results, asymptotic properties of distributions using wavelet transforms
$W_\psi f(b,a)=\langle \bar\psi(x), f(b+ax)\rangle \qquad(f\in{\mathcal S}'({\mathbb R}))$
that admit a reconstruction wavelet and defined by functions $$\psi\in {\mathcal S}_0({\mathbb R})$$.
A basic Tauberian proposition characterizes in terms of the behavior of such a wavelet $$W_\psi$$ at approaching points of the boundary the existence of a distribution $$g$$ defined by a quasi-asymptotic behavior
$\langle\varphi, g\rangle = \lim_{\varepsilon \downarrow 0} {1\over \varepsilon L(\varepsilon)} \langle f(x_0+\varepsilon x),\varphi(x)(x)\rangle,$
if $$f\in{\mathcal S}'_0({\mathbb R})$$ and $$L$$ is a given slowly varying function at $$0$$.
This proposition allows the authors to prove their main Tauberian theorems for quasi-asymptotics at points. Also corresponding Tauberian theorems for quasi-asymptotics at infinity are presented.
A number of clarifying examples and remarks are also included in this interesting paper.

### MSC:

 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 26A12 Rate of growth of functions, orders of infinity, slowly varying functions 40E05 Tauberian theorems 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 46F10 Operations with distributions and generalized functions
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