Discrete Convolution Associated with Fractional Cosine and Sine Series

Fractional sine series (FRSS) and fractional cosine series (FRCS) are the discrete form of the fractional cosine transform (FRCT) and fractional sine transform (FRST). The recent studies have shown that discrete convolution is widely used in optics, signal processing and applied mathematics. In this paper, firstly, the definitions of fractional sine series (FRSS) and fractional cosine series (FRCS) are presented. Secondly, the discrete convolution operations and convolution theorems for fractional sine and cosine series are given. The relationship of two convolution operations is presented. Lastly, the discrete Young’s type inequality is established. The proposed theory plays an important role in digital filtering and the solution of differential and integral equations.