A new strategy for choosing the Chebyshev‐gegenbauer parameters in a reconstruction based on asymptotic analysis

Abstract

The Gegenbauer reconstruction method, first proposed by Gottlieb et. al. in 1992, has been considered a useful technique for re‐expanding finite series polynomial approximations while simultaneously avoiding Gibbs artifacts. Since its introduction many studies have analyzed the method's strengths and weaknesses as well as suggesting several applications. However, until recently no attempts were made to optimize the reconstruction parameters, whose careful selection can make the difference between spectral accuracies and divergent error bounds.

In this paper we propose asymptotic analysis as a method for locating the optimal Gegenbauer reconstruction parameters. Such parameters are useful to applications of this reconstruction method that either seek to bound the number of Gegenbauer expansion coefficients or to control compression ratios. We then illustrate the effectiveness of our approach with the results from some numerical experiments.

Keywords:

• Chebyshev pseudo‐spectral approximation
• Gegenbauer reconstruction
• exponential convergence