Tauberian theorems for distributional wavelet transform

Abstract

In this paper we investigated the asymptotic behaviour at 0 and infinity of the distributional wavelet transform. Assuming that the wavelet transform 𝒲 _g f(b, a) has the ordinary asymptotic behaviour at 0 (resp. at infinity) with respect to both variables (resp. to the variable b), we obtained the result for the quasiasymptotic behaviour (resp. the S-asymptotics) at 0 (resp. at infinity) of the distribution f∈𝒮′(ℝ). Additionally, we proved that the distribution has the S-asymptotics at infinity equal to zero if its wavelet transform 𝒲 _g f(b, a) has the S-asymptotics at infinity with respect to the variable b.

Keywords:

• Wavelet transform
• Quasiasymptotic behaviour
• S-asymptotics

Acknowledgements

The authors are grateful to Professor Stevan Pilipović from University of Novi Sad for his guidance and comments during the preparation of this paper.