Improving Fourier partial sum approximation for discontinuous functions using a weight function. (English) Zbl 1470.42008

Yun, Beong In

Abstr. Appl. Anal. 2017, Article ID 1364914, 7 p. (2017).

Summary: We introduce a generalized sigmoidal transformation \(w_m(r; x)\) on a given interval \([a, b]\) with a threshold at \(x = r \in(a, b)\). Using \(w_m(r; x)\), we develop a weighted averaging method in order to improve Fourier partial sum approximation for a function having a jump-discontinuity. The method is based on the decomposition of the target function into the left-hand and the right-hand part extensions. The resultant approximate function is composed of the Fourier partial sums of each part extension. The pointwise convergence of the presented method and its availability for resolving Gibbs phenomenon are proved. The efficiency of the method is shown by some numerical examples.

MSC:

42A10

Trigonometric approximation

References:

[1]	Folland, G. B., Fourier analysis and Its applications, (1992), Pacific Grove, Calif, USA: Wadsworth & Brooks Cole Advanced Books & Software, Pacific Grove, Calif, USA · Zbl 0786.42001
[2]	Jerry, A. J., The Gibbs phenomenon in Fourier Analysis, Splines and Wavelet Approximations, (1998), London, UK: Kluwer Academic Publ, London, UK · Zbl 0918.42001
[3]	Boyd, J. P., Trouble with GEGenbauer reconstruction for defeating GIBbs’ phenomenon: Runge phenomenon in the diagonal limit of GEGenbauer polynomial approximations, Journal of Computational Physics, 204, 1, 253-264, (2005) · Zbl 1071.65189
[4]	Gelb, A.; Gottlieb, D., The resolution of the Gibbs phenomenon for spliced functions in one and two dimensions, Computers & Mathematics with Applications, 33, 11, 35-58, (1997) · Zbl 0911.65145
[5]	Gottlieb, D.; Shu, C.-W., On the Gibbs phenomenon and its resolution, SIAM Review, 39, 4, 644-668, (1997) · Zbl 0885.42003
[6]	Jung, J.-H.; Shizgal, B. D., Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomenon, Journal of Computational and Applied Mathematics, 172, 1, 131-151, (2004) · Zbl 1053.65102
[7]	Shizgal, B. D.; Jung, J.-H., Towards the resolution of the Gibbs phenomena, Journal of Computational and Applied Mathematics, 161, 1, 41-65, (2003) · Zbl 1033.65122
[8]	Jerri, A. J., Lanczos-like σ-factors for reducing the Gibbs phenomenon in general orthogonal expansions and other representations, Journal of Computational Analysis and Applications, 2, 2, 111-127, (2000) · Zbl 0952.42015
[9]	Tadmor, E.; Tanner, J., Adaptive mollifiers for high resolution recovery of piecewise smooth data from its spectral information, Foundations of Computational Mathematics, 2, 2, 155-189, (2002) · Zbl 1056.42002
[10]	Tadmor, E.; Tanner, J., Adaptive filters for piecewise smooth spectral data, IMA Journal of Numerical Analysis (IMAJNA), 25, 4, 635-647, (2005) · Zbl 1086.65124
[11]	Yun, B. I., A weighted averaging method for treating discontinuous spectral data, Applied Mathematics Letters, 25, 9, 1234-1239, (2012) · Zbl 1250.65164
[12]	Yun, B. I., A cumulative averaging method for piecewise polynomial approximation to discrete data, Applied Mathematical Sciences, 10, 5-8, 331-343, (2016)
[13]	Prössdorf, S.; Rathsfeld, A.; Petkov, V.; Lazarov, R., On an integral equation of the first kind arising from a cruciform crack problem, Integral Equations and Inverse Problems, (1991)
[14]	Elliott, D., Sigmoidal transformations and the trapezoidal rule, The Journal of the Australian Mathematical Society B: Applied Mathematics, 40, E77-E137, (1998) · Zbl 0928.65033