Fractional Mixed Weighted Convolution and Its Application in Convolution Integral Equations

The convolution integral equations are very important in optics and signal processing domain. In this paper, fractional mixed-weighted convolution is defined based on the fractional cosine transform; the corresponding convolution theorem is achieved. The properties of fractional mixed-weighted convolution and Young's type theorem are also explored. Based on the fractional mixed-weighted convolution and fractional cosine transform, two kinds of convolution integral equations are considered, the explicit solutions of fractional convolution integral equations are obtained, and the computational complexity of solutions are also analyzed.