Uncertainty principles for the generalized Fourier transform associated to a Dunkl-type operator

Abstract

In this paper, we prove various mathematical aspects of the qualitative and quantitative uncertainty principle, including Hardy’s, Cowling–Price and its variants, Beurling and its variants, Gelfand–Shilov, Donoho–Stark’s uncertainty principle, and variants of Heisenberg’s inequalities for the generalized Fourier transform associated to a Dunkl-type operator on the real line.

Keywords:

• generalized Fourier transform
• Hardy’s type theorem
• Cowling–Price’s theorem
• Beurling’s theorem
• Donoho–Stark’s uncertainty principle
• variants of Heisenberg’s uncertainty principle

AMS Subject Classifications:

• 42B10
• 42B35
• 43A32

View correction statement:

Erratum

Acknowledgements

The author is deeply indebted to the referees and K. Trimèche for their constructive comments and their helps in improving the contents of this paper. The author gratefully acknowledges the Deanship of Scientific Research at the Taibah University on the material and moral support.

Notes

No potential conflict of interest was reported by the author.

Dedicated to my Father Youssef Mejjaoli. This article was originally published with errors. This version has been corrected. Please see Erratum (http://dx.doi.org/10.1080/00036811.2015.1096121).