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 approximation of the points of discontinuities and the magnitudes of jumps of a 2π-periodic bounded function. Earlier, we used the derivative of the partial sums, while in this work we use integrals. First, we obtain new identities which determine the jumps of a 2π-periodic function of Vp, 1 ≤ p < 2, class, with a finite number of discontinuities, by means of the tails of its integrated Fourier series. Next, based on these identities we establish asymptotic expansions for the approximations of the location of the discontinuity and the magnitude of the jump of a 2π-periodic piecewise smooth function with one singularity. By an appropriate linear combination, obtained via integrals of different order, we significantly improve the accuracy of the initial approximations. Then, we apply Richardsonʹs extrapolation method to enhance the approximation results. For a function with multiple discontinuities we use simple formulae which “eliminate” all discontinuities of the function but one. Then we treat the function as if it had one singularity. Finally, we give the description of a programmable algorithm for the approximation of the discontinuities, investigate the stability of the method, study its complexity, and present some numerical results.