Determining Analyticity for Parameter Optimization of the Gegenbauer Reconstruction Method

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.