Spectral error bound in the mollification of Fourier approximation using Gegenbauer polynomial based mollifier

Abstract

Due to the global nature of the Fourier spectral methods, the Fourier approximation of discontinuous function gets impaired by spurious oscillations. This results in the approximation order reducing to and, respectively, for the discontinuous and non-discontinuous points. Nevertheless, it is shown by Gottlieb and others that higher order information is hidden in this approximation, using which spectral order accuracy can be recovered. Thus, in our work, we propose a spectral mollifier using the Gegenbauer polynomial kernel to regain the spectral accuracy of the Fourier approximation of a discontinuous function. Pointwise spectral accuracy has also been proved except for the discontinuous points. Results have been illustrated through examples. © The Author(s) under exclusive licence to Korean Society for Informatics and Computational Applied Mathematics 2025.