ABSTRACT
The hallmark of the fractional Laplacian is the relation between its Fourier transform and that of a function under consideration. This defining property provides us with a point of departure for extending concepts and definitions to trigonometric and other periodic functions in terms of Fourier series representations and eigenfunction expansions. This chapter presents fundamental stipulations regarding the fractional Laplacian of the sine and cosine functions. It defines the fractional Laplacian of a periodic function in terms of its Fourier series expansion. Having established the periodic differentiation matrix, the chapter shows how to develop a numerical method for solving the fractional Poisson equation for a periodic function.
