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

Abstract

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.

Fractional Mixed Weighted Convolution and Its Application in Convolution Integral Equations

“…K. Razminia and A. Razminia [21] studied fractional difusion equation (FDE) using the convolution integral. Li et al [22] analyzed the solvability of the convolution equations by convolution operator for a twodimensional fractional Fourier transform in polar coordinates. Feng and Wang [23] discussed explicit solutions of the convolution-type integral equations using the generalized fractional convolution.…”

The two-dimensional OLCT of angularly periodic functions in polar coordinates

“…It is a time-shifted and frequency-modulated version of LCT [11][12][13]. Many linear transforms widely used in practical applications, such as Fourier transform (FT), offset FT [7,9], fractional Fourier transform (FRFT) [7,9,14], offset FRFT [12,15], Fresnel transform (FRST) [2], LCT, etc., are all special cases of the OLCT. Therefore, studying the OLCT and developing related theories of the OLCT may help to gain more insights into its special situation and transfer the knowledge gained from one discipline to other disciplines.…”