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

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- by George Kvernadze;
- Math. Comp.
**73**(2004), 731-751 - DOI: https://doi.org/10.1090/S0025-5718-03-01594-1
- Published electronically: July 29, 2003
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## 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, MA, 1965. MR**197234** - 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, RI, 1967. MR**310533** - 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

## Bibliographic Information

**George Kvernadze**- Affiliation: Department of Mathematics, Weber State University, Ogden, Utah 84408
- Email: gkvernadze@weber.edu
- Received by editor(s): November 30, 2001
- Received by editor(s) in revised form: November 21, 2002
- Published electronically: July 29, 2003
- © Copyright 2003 American Mathematical Society
- Journal: Math. Comp.
**73**(2004), 731-751 - MSC (2000): Primary 65D99, 65T99, 42C10
- DOI: https://doi.org/10.1090/S0025-5718-03-01594-1
- MathSciNet review: 2031403