Analogues of Beurling's theorem for some integral transforms

Abstract

A famous result of Beurling says that if the Fourier transform of a non-zero integrable function on the real line has certain exponential decay, then the function cannot vanish on a set of positive Lebesgue measure. We prove an analogue of Beurling's theorem for Hankel transform and some several variable analogues of Beurling's theorem for Fourier transform as a consequence of similar results for Dunkl transform. We also prove an analogue of Beurling's theorem for spectral projections associated to the Dunkl-Laplacian.

Keywords:

• Hankel transform
• Dunkl transform
• Beurling's theorem
• harmonic measure

AMS Subject Classifications:

• Primary: 42B10
• Secondary: 44A05
• 30C85

Acknowledgments

The authors are grateful to the reviewer for careful reading of the manuscript and for very useful suggestions which helped in improving the same. The authors also wish to thank Prof. Swagato K. Ray and Dr. Mithun Bhowmik for several fruitful discussions and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author is supported by Council of Scientific and Industrial Research (CSIR), India (File No: 09/028(1002)/2017-EMR-I).