The Solvability of a Class of Convolution Equations Associated with 2D FRFT

In this paper, the solvability of a class of convolution equations is discussed by using two-dimensional (2D) fractional Fourier transform (FRFT) in polar coordinates. Firstly, we generalize the 2D FRFT to the polar coordinates setting. The relationship between 2D FRFT and fractional Hankel transform (FRHT) is derived. Secondly, the spatial shift and multiplication theorems for 2D FRFT are proposed by using this relationship. Thirdly, in order to analyze the solvability of the convolution equations, a novel convolution operator for 2D FRFT is proposed, and the corresponding convolution theorem is investigated. Finally, based on the proposed theorems, the solvability of the convolution equations is studied.

Boundedness of pseudo-differential operator associated with fractional Hankel transform

A pseudo-differential operator (p.d.o.) associated with the fractional Hankel transform involving the symbol a(x, ξ) is defined. An integral representation of p.d.o. and boundedness result of the composition of operators Δμr and A μ,α are obtained. A generalized integral operator A μ,aα corresponding to p.d.o. is also defined and the properties of the product of two generalized integral operators corresponding to p.d.o. are studied.