Journal of Fourier Analysis and Applications

, Volume 17, Issue 1, pp 65-95

First online:

Tauberian Theorems for the Wavelet Transform

  • Jasson VindasAffiliated withDepartment of Mathematics, Ghent University
  • , Stevan PilipovićAffiliated withDepartment of Mathematics and Informatics, University of Novi Sad Email author 
  • , Dušan RakićAffiliated withFaculty of Technology, University of Novi Sad

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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.


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

Mathematics Subject Classification (2000)

42C40 26A12 40E05 41A60 46F10 42F12