Tauberian theorems for the wavelet transform (Q628713)

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.