Detecting the singularities of a function of Vp class by its integrated Fourier series

In the present paper, we pursue the general idea suggested in our previous work. Namely, we utilize the truncated Fourier series as a tool for the app roximation of the points of discontinuities and the magnitudes of jumps of a 2 pi-periodic bounded function. Earlier, we used the derivative of the pa rtial sums, while in this work we use integrals. First, we obtain new identities which determine the jumps of a 2 pi-periodi c function of V-p, 1 less than or equal to p < 2, class, with a finite numb er of discontinuities, by means of the tails of its integrated Fourier seri es. Next, based on these identities we establish asymptotic expansions for the approximations of the location of the discontinuity and the magnitude of th e jump of a 2 pi-periodic piecewise smooth function with one singularity. B y an appropriate linear combination, obtained via integrals of different or der, we significantly improve the accuracy of the initial approximations. T hen, we apply Richardson's extrapolation method to enhance the approximatio n results. For a function with multiple discontinuities we use simple formu lae which "eliminate" all discontinuities of the function but one. Then we Great the function as if it had one singularity. Finally, we give the description of a programmable algorithm for the approx imation of the discontinuities, investigate the stability of the method, st udy its complexity, and present some numerical results. (C) 2000 Elsevier S cience Ltd. All rights reserved.