On uncertainty bounds and growth estimates for fractional fourier transforms

Abstract

Gelfand–Shilov spaces are spaces of entire functions defined in terms of a rate of growth in one direction and a concomitant rate of decay in an orthogonal direction. Gelfand and Shilov proved that the Fourier transform is an isomorphism among certain of these spaces. In this article we consider mapping properties of fractional Fourier transforms on Gelfand–Shilov spaces. Just as the Fourier transform corresponds to rotation through a right angle in the phase plane, fractional Fourier transforms correspond to rotations through intermediate angles. Therefore, the aim of fractional Fourier estimates is to set up a correspondence between growth properties in the complex plane versus decay properties in phase space.

Keywords:

• Hardy's theorem
• Beurling's theorem
• Gelfand–Shilov space
• Fractional fourier transform

Keywords:

• AMS 2000 Subject Classifications: Primary 42B10
• 42B35
• Secondary 30D15

Acknowledgments

The authors would like to thank John Benedetto for the helpful comments and Parag Mehta for discussions related to this work. The second author gratefully acknowledges the hospitality and support of Macquarie University through a MURG grant.