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The fractional Hilbert transform was introduced by Zayed [30, Zayed, 1998]and has been widely used in signal processing. In view of is connection withthe fractional Fourier transform, Chen, the first, second and fourth authors ofthis paper in [6, Chen et al., 2021] studied the fractional Hilbert transformand other fractional multiplier operators on the real line. The present paperis concerned with a natural extension of the fractional Hilbert transform tohigher dimensions: this extension is the fractional Riesz transform which isdefined by multiplication which a suitable chirp function on the fractionalFourier transform side. In addition to a thorough study of the fractional Riesztransforms, in this work we also investigate the boundedness of singularintegral operators with chirp functions on rotation invariant spaces, chirpHardy spaces and their relation to chirp BMO spaces, as well as applications ofthe theory of fractional multipliers in partial differential equations. Throughnumerical simulation, we provide physical and geometric interpretations ofhigh-dimensional fractional multipliers. Finally, we present an application ofthe fractional Riesz transforms in edge detection which verifies a hypothesisinsinuated in [26, Xu et al., 2016]. In fact our numerical implementationconfirms that amplitude, phase, and direction information can be simultaneouslyextracted by controlling the order of the fractional Riesz transform.

Riesz transform associated with the fractional Fourier transform and applications in image edge detection