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
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by George Kvernadze;

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

DOI: https://doi.org/10.1090/S0025-5718-10-02366-5

Published electronically: April 21, 2010

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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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