Approximation of the discontinuities of a function by its classical orthogonal polynomial Fourier coefficients

Kvernadze, George

doi:10.1090/S0025-5718-10-02366-5

Approximation of the discontinuities of a function by its classical orthogonal polynomial Fourier coefficients
HTML articles powered by AMS MathViewer

by George Kvernadze PDF

Math. Comp. 79 (2010), 2265-2285 Request permission

Abstract:

In the present paper, we generalize the method suggested in an earlier paper by the author and overcome its main deficiency.

First, we modify the well-known Prony method, which subsequently will be utilized for recovering exactly the locations of jump discontinuities and the associated jumps of a piecewise constant function by means of its Fourier coefficients with respect to any system of the classical orthogonal polynomials.

Next, we will show that the method is applicable to a wider class of functions, namely, to the class of piecewise smooth functions—for functions which piecewise belong to $C^2[-1,1]$, the locations of discontinuities are approximated to within $O(1/n)$ by means of their Fourier-Jacobi coefficients. Unlike the previous one, the generalized method is robust, since its success is independent of whether or not a location of the discontinuity coincides with a root of a classical orthogonal polynomial. In addition, the error estimate is uniform for any $[c,d]\subset (-1,1)$.

To the end, we discuss the accuracy, stability, and complexity of the method and present numerical examples.

References

V. M. Badkov, Convergence in the mean and almost everywhere of Fourier series in polynomials that are orthogonal on an interval, Mat. Sb. (N.S.) 95(137) (1974), 229–262, 327 (Russian). MR 0355464
Nana S. Banerjee and James F. Geer, Exponentially accurate approximations to periodic Lipschitz functions based on Fourier series partial sums, J. Sci. Comput. 13 (1998), no. 4, 419–460. MR 1676752, DOI 10.1023/A:1023289301743
R. B. Bauer, Numerical shock capturing techniques, Doctoral Thesis, Division of Applied Mathematics, Brown University, 1995.
Wei Cai, David Gottlieb, and Chi-Wang Shu, Essentially nonoscillatory spectral Fourier methods for shock wave calculations, Math. Comp. 52 (1989), no. 186, 389–410. MR 955749, DOI 10.1090/S0025-5718-1989-0955749-2
Knut S. Eckhoff, Accurate and efficient reconstruction of discontinuous functions from truncated series expansions, Math. Comp. 61 (1993), no. 204, 745–763. MR 1195430, DOI 10.1090/S0025-5718-1993-1195430-1
Knut S. Eckhoff, Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions, Math. Comp. 64 (1995), no. 210, 671–690. MR 1265014, DOI 10.1090/S0025-5718-1995-1265014-7
Knut S. Eckhoff, On a high order numerical method for functions with singularities, Math. Comp. 67 (1998), no. 223, 1063–1087. MR 1459387, DOI 10.1090/S0025-5718-98-00949-1
Walter Gautschi, Norm estimates for inverses of Vandermonde matrices, Numer. Math. 23 (1975), 337–347. MR 378382, DOI 10.1007/BF01438260
James Geer and Nana Saheb Banerjee, Exponentially accurate approximations to piece-wise smooth periodic functions, J. Sci. Comput. 12 (1997), no. 3, 253–287. MR 1600216, DOI 10.1023/A:1025649427614
Anne Gelb and Eitan Tadmor, Detection of edges in spectral data, Appl. Comput. Harmon. Anal. 7 (1999), no. 1, 101–135. MR 1699594, DOI 10.1006/acha.1999.0262
Anne Gelb and Eitan Tadmor, Detection of edges in spectral data. II. Nonlinear enhancement, SIAM J. Numer. Anal. 38 (2000), no. 4, 1389–1408. MR 1790039, DOI 10.1137/S0036142999359153
G. Hämmerlin and K. Hoffmann, Numerical analysis, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1991.
George Kvernadze, Determination of the jumps of a bounded function by its Fourier series, J. Approx. Theory 92 (1998), no. 2, 167–190. MR 1604919, DOI 10.1006/jath.1997.3125
George Kvernadze, Approximation of the singularities of a bounded function by the partial sums of its differentiated Fourier series, Appl. Comput. Harmon. Anal. 11 (2001), no. 3, 439–454. MR 1866350, DOI 10.1006/acha.2001.0362
George Kvernadze, Approximating the jump discontinuities of a function by its Fourier-Jacobi coefficients, Math. Comp. 73 (2004), no. 246, 731–751. MR 2031403, DOI 10.1090/S0025-5718-03-01594-1
H. N. Mhaskar and J. Prestin, On a build-up polynomial frame for the detection of singularities, Self-similar systems (Dubna, 1998) Joint Inst. Nuclear Res., Dubna, 1999, pp. 98–109. MR 1819426
H. N. Mhaskar and J. Prestin, Polynomial frames for the detection of singularities, Wavelet analysis and multiresolution methods (Urbana-Champaign, IL, 1999) Lecture Notes in Pure and Appl. Math., vol. 212, Dekker, New York, 2000, pp. 273–298. MR 1777997
H. N. Mhaskar and J. Prestin, On the detection of singularities of a periodic function, Adv. Comput. Math. 12 (2000), no. 2-3, 95–131. MR 1745108, DOI 10.1023/A:1018921319865
R. Prony, Essai experimental et analytique etc., J. L’Ecole Polytech. (Paris) 2 (1795), 24–76.
P. K. Suetin, Klassicheskie ortogonal′nye mnogochleny, “Nauka”, Moscow, 1979 (Russian). Second edition, augmented. MR 548727
Gábor Szegő, Orthogonal polynomials, 3rd ed., American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, R.I., 1967. MR 0310533
Daniel Waterman, On convergence of Fourier series of functions of generalized bounded variation, Studia Math. 44 (1972), 107–117. MR 310525, DOI 10.4064/sm-44-2-107-117