THE RESOLUTION OF THE GIBBS PHENOMENON FOR SPLICED FUNCTIONS IN ONE AND 2 DIMENSIONS

In this paper we study approximation methods for analytic functions th at have been ''spliced'' into nonintersecting subdomains. We assume th at we are given the first 2N + 1 Fourier coefficients for the function s in each subdomain. The objective is to approximate the ''spliced'' f unction in each subdomain and then to ''glue'' the approximations toge ther in order to recover the original function in the full domain. The Fourier partial sum approximation in each subdomain yields poor resul ts, as the convergence is slow and spurious oscillations occur at the boundaries of each subdomain. Thus once we ''glue'' the subdomain appr oximations back together, the approximation for the function in the fu ll domain will exhibit oscillations throughout the entire domain. Rece ntly methods have been developed that successfully eliminate the Gibbs phenomenon for analytic but nonperiodic functions in one dimension. T hese methods are based on the knowledge of the first 2N + 1 Fourier co efficients and use either the Gegenbauer polynomials (Gottlieb et al.) or the Bernoulli polynomials (Abarbanel, Gottlieb, Cai et al., and Ec khoff). We propose a way to accurately reconstruct a ''spliced'' funct ion in a full domain by extending the current methods to eliminate the Gibbs phenomenon in each nonintersecting subdomain and then ''gluing' ' the approximations back together. We solve this problem in both one and two dimensions. In the one-dimensional case we provide two alterna tive options, the Bernoulli method and the Gegenbauer method, as well as a new hybrid method, the Gegenbauer-Bernoulli method. In the tw-dim ensional case we prove, for the very first time, exponential convergen ce of the Gegenbauer method, and then we apply it to solve the ''splic ed'' function problem.