Convolution theorem for the windowed linear canonical transform

In this paper, we obtain the convolution theorems for the windowed linear canonical transform (WLCT). According to the WLCT of a convolution of two functions is the product of their respective WLCTs, the spectral convolution theorem of the WLCT is derived. Then, the spatial convolution theorem of the WLCT is exploited by a new spatial convolution operator. Moreover, applying the mathematical inequalities, the existence theorems of the convolution for the WLCT are established. Finally, as an application, the solution of a convolution equation is given by the convolution theorem of the WLCT.