Construction of Lanczos type filters for the Fourier series approximation

In order to improve the error such as that from the Gibbs phenomenon appearing in the truncated Fourier series approximation for discontinuous functions, we develop a new filtering method based on the sigmoidal transformation. The presented method results in a multiplicative factor, named Lanczos type sigmoidal filter (LSF), in the form of the Fourier transform of a derivative of a sigmoidal transformation. It can be seen that the sigmoidal filter is a generalization of the existing Lanczos filter. Particularly, employing some well known sigmoidal transformations, we derive closed forms of the sigmoidal filters. Moreover, we propose an asymptotically higher order filter which is competitive with an adaptive filter achieving exponential accuracy away from the discontinuity. By numerical experiment we show that the new filters are available for decreasing the rise time as well as resolving the Gibbs phenomenon of the truncated Fourier series approximation to discontinuous functions.