Fourier series of half-range functions by smooth extension

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.