Lanczos-Like σ-Factors for Reducing the Gibbs Phenomenon in General Orthogonal Expansions and Other Representations
Abdul J. Jerri · Journal of Computational Analysis and Applications · 2000
The first attempt for reducing the Gibbs phenomenon in an orthogonalexpansion, besides the usual one of Fourier series, is due to Cooke in1927–1928 for the Fourier Bessel series.However, his work was limited tothe well-known Fejer averaging of the series. For the past 10 years or so,we have tried a parallel to the more effective Lanczos-type “localaveraging method” of the Fourier series. As expected, such efforts werehindered by the lack of “realizable” tools for the general orthogonalexpansion that parallels the familiar simple ones of the Fourier seriesand transforms. During the past 3 years, we have succeeded in developinga simple direct method of filtering the Gibbs phenomenon in Fourier--Besselseries, Hankel transforms representation, and a number of orthogonalpolynomials series expansions. This parallels the equivalent result ofLanczos, which he obtained for his local averaging with the help ofthe (Fourier) convolution theorem.