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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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.

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