Abstract
In this paper we derive error estimates for two filters based on piecewise polynomial interpolations of zeroth and first degrees. For a piecewise smooth function f(x) in [0,1], we show that, if all the discontinuity points of f(x) are nodes then, using these filters, we can reconstruct point values of f(x) accurately even near discontinuity points. If f(x) is a piecewise constant or a linear function, the reconstruction formulas are exact. We also propose reconstruction formulas such that we can compute the (approximate ) point values of f(x) using the fast Fourier transform, even when using non-uniform meshes. Several numerical experiments are also provided to illustrate the results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Bertero, C. Demol and E.R. Pike, Linear inverse problems with discrete data, I: General formulation and singular system analysis, Inverse Problems 1 (1985) 301–330.
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods in Fluid Dynamics (Springer, Berlin, 1988).
W.K. Cheung and R.M. Lewitt, Modified Fourier reconstruction method using shifted Transform samples, Phys. Med. Biol. 36(2) (1991) 269–277.
A.R. De Pierro and M. Wei, Iterative methods to detect discontinuity points and recover point values of a function from its Fourier coefficients, submitted.
W. Gautschi, Attenuation factors in practical Fourier analysis, Numer. Math. 18 (1972) 373–400.
D. Gottlieb and S. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications (SIAM, Philadelphia, PA, 1977).
D. Gottlieb, C.W. Shu, A. Solomonoff and H. Vandeven, On the Gibbs phenomenon I: Recovering exponential accuracy from the Fourier partial sum of the a nonperiodic analytic function, J. Comput. Appl. Math. 43 (1992) 81–98.
D. Gottlieb and C.W. Shu, On the Gibbs phenomenon and its resolution, SIAM Rev. 39 (1997) 644–668.
M.H. Gutknecht, Attenuation factors in multivariate Fourier analysis, Numer. Math. 51 (1987) 615–629.
G. Hammerlin and K.-H. Hoffmann, Numerical Mathematics (Springer, Berlin, 1991).
G. Hamming, Numerical Methods for Scientists and Engineers, 2nd ed. (McGraw-Hill, New York, 1973).
A.K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, NJ, 1989).
A.C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
F. Locher, Interpolation on uniform meshes by the translates of one function and related attenuation factors, Math. Comp. 37(156) (1981) 403–416.
F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1985).
D.H. Vandeven, Family of spectral filters for discontinuous problem, J. Sci. Comput. 6 (1991) 159–192.
J. Yin, A. De Pierro and M. Wei, Reconstruction of a compactly supported function from the discrete sampling of its Fourier transform, IEEE Trans. Signal Process. 47 (1999) 3356–3364.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wei, M., De Pierro, A.R. & Yin, J. Error Estimates for Two Filters Based on Polynomial Interpolation for Recovering a Function from Its Fourier Coefficients. Numerical Algorithms 35, 205–231 (2004). https://doi.org/10.1023/B:NUMA.0000021769.62129.2d
Issue Date:
DOI: https://doi.org/10.1023/B:NUMA.0000021769.62129.2d