Riesz transform associated with the fractional Fourier transform and applications

Since Zayed \cite[Zayed, 1998]{z2} introduced the fractional Hilbert transform related to the fractional Fourier transform, this transform has been widely concerned and applied in the field of signal processing. Recently, Chen, the first, second and fourth authors \cite[Chen et al, 2021]{cfgw} attribute it to the operator corresponding to fractional multiplier, but it is only limited to 1-dimensional case. This paper naturally considers the high-dimensional situation. We introduce the fractional Riesz transform associated with fractional Fourier transform, in which the chirp function is the key factor and the technical barriers to be overcome. Furthermore, after equipping with chirp functions, we introduce and investigate the boundedness of singular integral operators, the dual properties of Hardy spaces and BMO spaces as well as the applications of theory of fractional multiplier in partial differential equation, which completely matched some classical results. Through numerical simulation, we give the physical and geometric interpretation of the high-dimensional fractional multiplier theorem. Finally, we present the application of the fractional Riesz transform in edge detection, which verifies the prediction proposed in \cite[Xu et al, 2016]{xxwqwy}. Moreover, the application presented in this paper can also be considered as the high-dimensional case of the application of the continuous fractional Hilbert transform in edge detection in \cite[Pei and Yeh, 2000]{py}.