Locating Discontinuities of a Bounded Function by the Partial Sums of Its Fourier Series

Authors:
George Kvernadze;Thomas Hagstrom;Henry Shapiro
Affiliations:
Department of Mathematics, Weber State University, Ogden, Utah 84408. gkvernadze@weber.edu;Department of Mathematics and Statistics, The University of New Mexico, Albuquerque, New Mexico 87131. hagstrorm@math.unm.edu;Department of Computer Science, The University of New Mexico, Albuquerque, New Mexico 87131. shapiro@cs.unm.edu
Venue:
Journal of Scientific Computing
Year:
1999

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Abstract

A key step for some methods dealing with the reconstruction of a function with jump discontinuities is the accurate approximation of the jumps and their locations. Various methods have been suggested in the literature to obtain this valuable information. In the present paper, we develop an algorithm based on asymptotic expansion formulae obtained in our earlier work. The algorithm enables one to approximate the locations of discontinuities and the magnitudes of jumps of a bounded function given its truncated Fourier series. We investigate the stability of the method and study its complexity. Finally, we consider several numerical examples in order to emphasize strong and weak points of the algorithm.