TY - JOUR

T1 - The resolution of the Gibbs phenomenon for "spliced" functions in one and two dimensions

AU - Gelb, A.

AU - Gottlieb, D.

N1 - Funding Information:
Consider an analytic and periodic function f(x) defined on \[a, b\] that has been split into two nonintersecting functions fl(x) and f2(x) in the subdomains \[a,c\] and \[c, b\], where a < c < b. The functions in each subdomain are no longer periodic. Now, suppose that the first 2N + 1 The authors were supported by AFOSR Grant F49620-96-1-0150 and NSF Grant DMS 95008. The first author was also supported by the National Science Foundation under Cooperative Agreement No. 9120008.

PY - 1997/6

Y1 - 1997/6

N2 - In this paper we study approximation methods for analytic functions that have been "spliced" into nonintersecting subdomains. We assume that we are given the first 2N + 1 Fourier coefficients for the functions in each subdomain. The objective is to approximate the "spliced" function in each subdomain 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 subdomain yields poor results, as the convergence is slow and spurious oscillations occur at the boundaries of each subdomain. Thus once we "glue" the subdomain 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 nonperiodic functions in one dimension. These methods are based on the knowledge of the first 2N + 1 Fourier coefficients and use either the Gegenbauer polynomials (Gottlieb et al.) or the Bernoulli polynomials (Abarbanel, Gottlieb, Cai et al., and Eckhoff). We propose a way to accurately reconstruct a "spliced" function 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 alternative options, the Bernoulli method and the Gegenbauer method, as well as a new hybrid method, the Gegenbauer-Bernoulli method. In the two-dimensional case we prove, for the very first time, exponential convergence of the Gegenbauer method, and then we apply it to solve the "spliced" function problem.

AB - In this paper we study approximation methods for analytic functions that have been "spliced" into nonintersecting subdomains. We assume that we are given the first 2N + 1 Fourier coefficients for the functions in each subdomain. The objective is to approximate the "spliced" function in each subdomain 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 subdomain yields poor results, as the convergence is slow and spurious oscillations occur at the boundaries of each subdomain. Thus once we "glue" the subdomain 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 nonperiodic functions in one dimension. These methods are based on the knowledge of the first 2N + 1 Fourier coefficients and use either the Gegenbauer polynomials (Gottlieb et al.) or the Bernoulli polynomials (Abarbanel, Gottlieb, Cai et al., and Eckhoff). We propose a way to accurately reconstruct a "spliced" function 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 alternative options, the Bernoulli method and the Gegenbauer method, as well as a new hybrid method, the Gegenbauer-Bernoulli method. In the two-dimensional case we prove, for the very first time, exponential convergence of the Gegenbauer method, and then we apply it to solve the "spliced" function problem.

KW - Bernoulli polynomials

KW - Exponential accuracy

KW - Fourier series

KW - Gegenbauer polynomials

KW - Gibbs phenomenon

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U2 - 10.1016/S0898-1221(97)00086-2

DO - 10.1016/S0898-1221(97)00086-2

M3 - Article

AN - SCOPUS:0031167838

VL - 33

SP - 35

EP - 58

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 11

ER -