On the fractional space-time Fourier transforms

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.