An averaging method for the Fourier approximation to discontinuous functions

Both the truncated Fourier integral and the truncated Fourier series approximations for a discontinuous function bring about the inevitable oscillating error, say, the Gibbs phenomenon. Most basic filtering methods for the Gibbs phenomenon like the Fejer averaging method and the Lanczos averaging method have a disadvantage that the rise time is very slow even though the filtering effect is prominent away from the discontinuity. In this paper, we propose a new averaging method of polynomial type which improves the rise time of the existing method. The present method can be regarded as a generalization of the traditional Lanczos averaging method. By several numerical examples, we show the efficiency of the present method.