Approximation of the Singularities of a Bounded Function by the Partial Sums of Its Differentiated Fourier Series

The problem of locating the discontinuities of a function by means of its truncated Fourier series, an interesting question in and of itself, arises naturally from an attempt to overcome the Gibbs phenomenon: the poor approximative properties of the Fourier partial sums of a discontinuous function. In [3], Cai et al. developed the idea already introduced in their previous papers and suggested a method for the reconstruction of a discontinuous function from the partial sums of its Fourier series. A key step of the method is the accurate approximation of the locations of singularities and the magnitudes of jumps of the function. Namely, let g be a 2π -periodic function with a finite number, M , of jump discontinuities that is piecewise smooth on the period. In addition, let us assume that the first 2n+ 1 Fourier coefficients of the function are known. If G(θ)= (π − θ)/2, θ ∈ (0,2π), is the 2π -periodic sawtooth function, then the function g can be represented as