Spline-Fourier Approximations of Discontinuous Waves

In the Fourier series approximation of real functions discontinuities of the functions or their derivatives cause problems like Gibbs phenomenon or slow uniform convergence. In the case of a nite number of isolated discontinuities the problems can be to a large extend recti ed by using periodic splines in the series. This modi ed Fourier series (Spline-Fourier series) is applied to the numerical solution of the wave equation (in periodic form) where discontinuities in the data functions or their derivatives appear quite often. The solution is sought in the form of a Spline-Fourier series about the space variable and close bounds are obtained using a certain iterative procedure of Newton type.