Determining Analyticity for Parameter Optimization of the Gegenbauer Reconstruction Method

Authors:
A. Gelb;Z. Jackiewicz
Affiliations:
-;-
Venue:
SIAM Journal on Scientific Computing
Year:
2005

Citing 0
Cited 4

Trouble with Gegenbauer reconstruction for defeating Gibbs' phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations

Journal of Computational Physics
Short note: Padé-Gegenbauer suppression of Runge phenomenon in the diagonal limit of Gegenbauer approximations

Journal of Computational Physics
Reconstruction of Piecewise Smooth Functions from Non-uniform Grid Point Data

Journal of Scientific Computing
Optimal Gegenbauer quadrature over arbitrary integration nodes

Journal of Computational and Applied Mathematics

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Abstract

The Gegenbauer reconstruction method effectively eliminates the Gibbs phenomenon and restores exponential accuracy to the approximations of piecewise smooth functions. Recent investigations show that its success depends upon choosing parameters in such a way that the regularization and the truncation error estimates are equally considered. This paper shows that the underlying analyticity of the function in smooth regions plays a critical role in the regularization error estimate. Hence we develop a technique that first analyzes the behavior of the function in its regions of smoothness and then applies this knowledge to refine the regularization error estimate. Such refinement yields better parameter choices for the Gegenbauer reconstruction method, and is confirmed both by better accuracy and more robustness in the approximation of piecewise smooth functions.