Approximating the jump discontinuities of a function by its Fourier-Jacobi coefficients

Kvernadze, George

doi:10.1090/S0025-5718-03-01594-1

Approximating the jump discontinuities of a function by its Fourier-Jacobi coefficients
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Math. Comp. 73 (2004), 731-751 Request permission

Abstract:

In the present paper we generalize Eckhoff’s method, i.e., the method for approximating the locations of discontinuities and the associated jumps of a piecewise smooth function by means of its Fourier-Chebyshev coefficients.

A new method enables us to approximate the locations of discontinuities and the associated jumps of a discontinuous function, which belongs to a restricted class of the piecewise smooth functions, by means of its Fourier-Jacobi coefficients for arbitrary indices. Approximations to the locations of discontinuities and the associated jumps are found as solutions of algebraic equations. It is shown as well that the locations of discontinuities and the associated jumps are recovered exactly for piecewise constant functions with a finite number of discontinuities.

In addition, we study the accuracy of the approximations and present some numerical examples.

References

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,” Doctor. 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
Günther Hämmerlin and Karl-Heinz Hoffmann, Numerical mathematics, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1991. Translated from the German by Larry Schumaker; Readings in Mathematics. MR 1088482, DOI 10.1007/978-1-4612-4442-4
Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1990. Corrected reprint of the 1985 original. MR 1084815
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
G. Kvernadze, T. Hagstrom, and H. Shapiro, Locating discontinuities of a bounded function by the partial sums of its Fourier series, J. Sci. Comput. 4 (1999), 301-327.
G. Kvernadze, T. Hagstrom, and H. Shapiro, Detecting the singularities of a function of $V_p$ class by its integrated Fourier series, Comput. Math. Appl. 39 (2000), no. 9-10, 25–43. MR 1753559, DOI 10.1016/S0898-1221(00)00084-5
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
Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
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), 95-131.
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
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