Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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


Author: George Kvernadze
Journal: Math. Comp. 79 (2010), 2265-2285
MSC (2010): Primary 65D99, 65T99, 42C10
DOI: https://doi.org/10.1090/S0025-5718-10-02366-5
Published electronically: April 21, 2010
MathSciNet review: 2684364
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. V. M. Badkov, Convergence in the mean and almost everywhere of Fourier series in polynomials orthogonal on an interval, Math. USSR Sbornik 2 (1974), 223-256. MR 0355464 (50:7938)
  • 2. N. S. Banerjee and J. F. Geer, Exponentially accurate approximations to periodic Lipschitz functions based on Fourier series partial sums, J. Sci. Comput. 13 (1998), 419-460. MR 1676752 (2000b:65020)
  • 3. R. B. Bauer, Numerical shock capturing techniques, Doctoral Thesis, Division of Applied Mathematics, Brown University, 1995.
  • 4. W. Cai, D. Gottlieb, and C.-W. Shu, Essentially nonoscillatory spectral Fourier methods for shock wave calculations, Math. Comp. 52 (1989), 389-410. MR 955749 (90a:65212)
  • 5. K. S. Eckhoff, Accurate and efficient reconstruction of discontinuous functions from truncated series expansions, Math. Comp. 61 (1993), 745-763. MR 1195430 (94a:65073)
  • 6. K. S. Eckhoff, Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions, Math. Comp. 64 (1995), 671-690. MR 1265014 (95f:65234)
  • 7. K. S. Eckhoff, On a high order numerical method for functions with singularities, Math. Comp. 67 (1998), 1063-1087. MR 1459387 (98j:65014)
  • 8. W. Gautschi, Norm estimates for inverses of Vandermonde matrices, Numer. Math. 23 (1975), 337-347. MR 0378382 (51:14550)
  • 9. J. Geer and N. S. Banerjee, Exponentially accurate approximations to piece-wise smooth periodic functions, J. Sci. Comput. 12 (1997), 253-287. MR 1600216 (98m:41032)
  • 10. A. Gelb and E. Tadmor, Detection of edges in spectral data, Appl. Comput. Harmon. Anal. 7 (1999), 101-135. MR 1699594 (2000g:42003)
  • 11. A. Gelb and E. Tadmor, Detection of edges in spectral data II. Nonlinear enhancement, SIAM J. Numer. Anal. 38 (2000), 1389-1408. MR 1790039 (2001i:42003)
  • 12. G. Hämmerlin and K. Hoffmann, Numerical analysis, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1991.
  • 13. G. Kvernadze, Determination of the jumps of a bounded function by its Fourier series , J. of Approx. Theory 92 (1998), 167-190. MR 1604919 (99m:42005)
  • 14. G. Kvernadze, Approximation of the singularities of a bounded function by the partial sums of its differentiated Fourier series, Appl. Comput. Harmon. Anal. 11 (2001), 439-454. MR 1866350 (2002i:42001)
  • 15. G. Kvernadze, Approximating the jump discontinuities of a function by its Fourier-Jacobi coefficients, Math. Comp. 73 (2004), 731-751. MR 2031403 (2004j:42026)
  • 16. H. N. Mhaskar and J. Prestin, On a build-up polynomial frame for the detection of singularities, in Self-Similar Systems, V. B. Priezzhev and V. P. Spiridonov, eds., Joint Institute for Nuclear Research, Dubna, Russia, 1999, pp. 98-109. MR 1819426 (2003b:94009)
  • 17. H. N. Mhaskar and J. Prestin, Polynomial frames for the detection of singularities, in Wavelet Analysis and Multiresolution Methods, Tian-Xiao He, ed., Lecture Notes in Pure and Applied Mathematics, Vol. 212, Marcel Decker, 2000, pp. 273-298. MR 1777997 (2001k:65212)
  • 18. H. N. Mhaskar and J. Prestin, On the detection of singularities of a periodic function, Adv. Comput. Math. 12 (2000), 95-131. MR 1745108 (2001a:42003)
  • 19. R. Prony, Essai experimental et analytique etc., J. L'Ecole Polytech. (Paris) 2 (1795), 24-76.
  • 20. P. K. Suetin, Classical orthogonal polynomials, 2nd rev. ed., Nauka, Moscow, 1979. MR 548727 (80h:33001)
  • 21. G. Szegő, Orthogonal polynomials, 3rd ed., Amer. Math. Soc. Colloq. Publ., Vol. 23, Amer. Math. Soc., Providence, RI, 1967. MR 0310533 (46:9631)
  • 22. D. Waterman, On convergence of Fourier series of functions of generalized bounded variation, Studia Math. 44 (1972), 107-117. MR 0310525 (46:9623)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65D99, 65T99, 42C10

Retrieve articles in all journals with MSC (2010): 65D99, 65T99, 42C10


Additional Information

George Kvernadze
Affiliation: Department of Mathematics, Weber State University, Ogden, Utah 84408
Email: gkvernadze@weber.edu

DOI: https://doi.org/10.1090/S0025-5718-10-02366-5
Keywords: The classical orthogonal polynomials, Fourier coefficients, piecewise smooth functions, approximation of discontinuities
Received by editor(s): November 23, 2007
Received by editor(s) in revised form: January 19, 2009, and June 20, 2009
Published electronically: April 21, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society