In this thesis we mainly focus on two main topics: the first one is to detect the location and recover the amplitude of the edges (jump discontinuities) of one-dimensional underlying functions that are compactly supported, L₂ functions and smooth except for finitely many jump discontinuities, and analyze the approximation error of our method of the edge detector by the concentration factor method from the only given spectral information, that is, the finitely many non-uniform samples of their Fourier data in frequency space ; the second is to reconstruct the piecewise smooth function, and analyze the approximation error of our reconstruction method by the hybrid filter-extrapolation method from the functions' spectral data.

In the first part of this thesis, to detect edges of the piecewise smooth function, we present a mathematical form of a proposed method of edge detector based on concentration factors. It is a vigorously computational approach of the edge points. We also present some admissible conditions for the concentration factor in order to achieve good convergent result of our reconstruction of the edge points.

While recovering functions, spectral projections enjoy superior resolution provided the data is globally smooth, while the presence of jump discontinuities is responsible for Gibbs phenomenon in the neighborhood of edges and an overall deterioration to the unacceptable 1st-order convergence rate. The purpose of the second part is to gain the superior accuracy in the piecewise smooth case. We will achieve this by using non-compactly supported filters away from jump discontinuities and stable extrapolation near jump discontinuities. Based on our error decompositions, we obtain the uniform exponential accuracy over the whole region.