On the fractional space-time Fourier transforms

ABSTRACT

The notion of a fractional space-time Fourier transform (FSFT) is outlined in this paper, and the properties of invertibility, linearity, Plancherel and others are derived. By establishing the relationship between the FSFT and space-time Fourier transform, a directional uncertainty principle (UP) and its specialization to coordinates are proved. Moreover, the Hardy UP of the FSFT is also obtained. The fractional Mustard convolution for space-time valued signals is introduced and written in terms of the standard convolution. Finally, the FSFT is employed in solving a partial differential equation in space-time analysis.

KEYWORDS:

• Space-time algebra
• space-time Fourier transform
• fractional Fourier transform
• uncertainty principles
• convolution
• Mustard convolution

AMS CLASSIFICATIONS:

• Primary 15A66
• 42B10
• Secondary 30G35
• 44A05
• 44A35

Acknowledgements

The authors would like to express their sincere gratitude to the editor and the two anonymous reviewers for their valuable comments and dedicated efforts in reviewing the paper, which have greatly contributed to its improvement. The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through large group Research Project under grant number RGP.2/432/44.

Disclosure statement

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

Additional information

Funding

This work is funded through large group Research Project [grant number RGP.2/432/44] provided by the Deanship of Scientific Research at King Khalid University, Saudi Arabia.