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 | 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 , the locations of discontinuities are approximated to within 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 .

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.