Fourier series of half-range functions by smooth extension

Abstract

This paper considers Fourier series approximations of one- and two-dimensional functions over the half-range, that is, over the sub-interval [0, L ] of the interval [− L, L ] in one-dimensional problems and over the sub-domain [0, L x ] × [0, L y ] of the domain [− L x , L x ] × [− L y , L y ] in two-dimensional problems. It is shown how to represent these functions using a Fourier series that employs a smooth extension. The purpose of the smooth extension is to improve the convergence characteristics otherwise obtained using the even and odd extensions. Significantly improved convergence characteristics are illustrated in one-dimensional and two-dimensional problems.