Journal of Fourier Analysis and Applications

, Volume 17, Issue 1, pp 65–95 | Cite as

Tauberian Theorems for the Wavelet Transform

Article

Abstract

We make a complete wavelet analysis of asymptotic properties of distributions. The study is carried out via Abelian and Tauberian type results, connecting the boundary asymptotic behavior of the wavelet transform with local and non-local quasiasymptotic properties of elements in the Schwartz class of tempered distributions. Our Tauberian theorems are full characterizations of such asymptotic properties. We also provide precise wavelet characterizations of the asymptotic behavior of elements in the dual of the space of highly time-frequency localized functions over the real line. For the use of the wavelet transform in local analysis, we study the problem of extensions of distributions initially defined on ℝ∖{0} to ℝ; in this extension problem, we explore the asymptotic properties of extensions of a distribution having a prescribed asymptotic behavior. Our results imply intrinsic properties of functions and measures as well, for example, we give a new proof of the classical Littlewood Tauberian theorem for power series.

Keywords

Wavelet transform Abelian theorems Tauberian theorems Inverse theorems Distributions Quasiasymptotics Slowly varying functions 

Mathematics Subject Classification (2000)

42C40 26A12 40E05 41A60 46F10 42F12 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Jasson Vindas
    • 1
  • Stevan Pilipović
    • 2
  • Dušan Rakić
    • 3
  1. 1.Department of MathematicsGhent UniversityGentBelgium
  2. 2.Department of Mathematics and InformaticsUniversity of Novi SadNovi SadSerbia
  3. 3.Faculty of TechnologyUniversity of Novi SadNovi SadSerbia

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