The Resolution Of The Gibbs Phenomenon For "Spliced" Functions In One And Two Dimensions

In this paper we study approximation methods for analytic functions that have been "spliced" into non-intersecting sub-domains. We assume that we are given the first 2N + 1 Fourier coefficients for the functions in each sub-domain. The objective is to approximate the "spliced" function in each sub-domain and then to "glue" the approximations together in order to recover the original function in the full domain. The Fourier partial sum approximation in each sub-domain yields poor results, as the convergence is slow and spurious oscillations occur at the boundaries of each sub-domain. Thus once we "glue" the sub-domain approximations back together, the approximation for the function in the full domain will exhibit oscillations throughout the entire domain. Recently methods have been developed that successfully eliminate the Gibbs phenomenon for analytic but non-periodic functions in one dimension. These methods are based on the knowledge of the first 2N + 1 Fourier coefficients and use ..