Spectral reconstruction of piecewise smooth functions from their discrete data
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 2, p. 155-175

This paper addresses the recovery of piecewise smooth functions from their discrete data. Reconstruction methods using both pseudo-spectral coefficients and physical space interpolants have been discussed extensively in the literature, and it is clear that an a priori knowledge of the jump discontinuity location is essential for any reconstruction technique to yield spectrally accurate results with high resolution near the discontinuities. Hence detection of the jump discontinuities is critical for all methods. Here we formulate a new localized reconstruction method adapted from the method developed in Gottlieb and Tadmor (1985) and recently revisited in Tadmor and Tanner (in press). Our procedure incorporates the detection of edges into the reconstruction technique. The method is robust and highly accurate, yielding spectral accuracy up to a small neighborhood of the jump discontinuities. Results are shown in one and two dimensions.

DOI : https://doi.org/10.1051/m2an:2002008
Classification:  42A10,  42A50,  65T40
Keywords: edge detection, nonlinear enhancement, concentration method, piecewise smoothness, localized reconstruction
@article{M2AN_2002__36_2_155_0,
author = {Gelb, Anne and Tadmor, Eitan},
title = {Spectral reconstruction of piecewise smooth functions from their discrete data},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {36},
number = {2},
year = {2002},
pages = {155-175},
doi = {10.1051/m2an:2002008},
zbl = {1056.42001},
mrnumber = {1906813},
language = {en},
url = {http://www.numdam.org/item/M2AN_2002__36_2_155_0}
}

Gelb, Anne; Tadmor, Eitan. Spectral reconstruction of piecewise smooth functions from their discrete data. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 2, pp. 155-175. doi : 10.1051/m2an:2002008. http://www.numdam.org/item/M2AN_2002__36_2_155_0/

[0] N.S. Banerjee and J. Geer, Exponential approximations using Fourier series partial sums, ICASE Report No. 97-56, NASA Langley Research Center (1997).

[0] N. Bary, Treatise of Trigonometric Series. The Macmillan Company, New York (1964). | MR 171116 | Zbl 0129.28002

[0] H.S. Carslaw, Introduction to the Theory of Fourier's Series and Integrals. Dover (1950).

[0] K.S. Eckhoff, Accurate reconstructions of functions of finite regularity from truncated series expansions. Math. Comp. 64 (1995) 671-690. | Zbl 0830.65144

[0] K.S. Eckhoff, On a high order numerical method for functions with singularities. Math. Comp. 67 (1998) 1063-1087. | Zbl 0895.65067

[0] A. Gelb and E. Tadmor, Detection of edges in spectral data. Appl. Comput. Harmon. Anal. 7 (1999) 101-135. | Zbl 0952.42001

[0] A. Gelb and E. Tadmor, Detection of edges in spectral data. II. Nonlinear Enhancement. SIAM J. Numer. Anal. 38 (2001) 1389-1408. | Zbl 0990.42025

[0] B.I. Golubov, Determination of the jump of a function of bounded $p$-variation by its Fourier series. Math. Notes 12 (1972) 444-449. | Zbl 0259.42007

[0] D. Gottlieb and C.-W. Shu, On the Gibbs phenomenon and its resolution. SIAM Rev. (1997). | MR 1491051 | Zbl 0885.42003

[0] D. Gottlieb and E. Tadmor, Recovering pointwise values of discontinuous data within spectral accuracy, in Progress and Supercomputing in Computational Fluid Dynamics, Proceedings of a 1984 U.S.-Israel Workshop, Progress in Scientific Computing, Vol. 6, E.M. Murman and S.S. Abarbanel Eds., Birkhauser, Boston (1985) 357-375. | Zbl 0597.65099

[0] G. Kvernadze, Determination of the jump of a bounded function by its Fourier series. J. Approx. Theory 92 (1998) 167-190. | Zbl 0902.42001

[0] E. Tadmor and J. Tanner, Adaptive mollifiers for high resolution recovery of piecewise smooth data from its spectral information, Foundations of Comput. Math. Online publication DOI: 10.1007/s002080010019 (2001), in press. | MR 1894374 | Zbl 1056.42002

[0] A. Zygmund, Trigonometric Series. Cambridge University Press (1959). | MR 107776 | Zbl 0085.05601