The discrete convolution for fractional cosine-sine series and its application in convolution equations

The fractional sine series (FRSS) and the fractional cosine series (FRCS) were defined. Three types of discrete convolution operations for FRCS and FRSS were introduced, along with a detailed investigation into their corresponding convolution theorems. The interrelationship between these convolution operations was also discussed. Additionally, as an application of the presented results, two forms of discrete convolution equations based on the proposed convolution theorems were examined, resulting in explicit solutions for these equations. Furthermore, numerical simulations were provided to demonstrate that our proposed solution can be easily implemented with low computational complexity.