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Mathematics of Computation

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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
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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.


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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.