On spectral estimation and Fourier transform approximation from sampled data

A method for spectral estimation of a continuous-domain signal, given by its sampled version only, is introduced. Unlike the discrete Fourier transform (DFT), the proposed approach reduces aliasing effects. The proposed approach relies on finite-duration Sobolev functions, for which the ideal sampling process is characterized by means of an inner product operation. The point-wise evaluation of the Fourier transform is based on a Sobolev type inner product too, allowing for a minimax approximation approach to be derived and utilized. Experimental results show that the proposed approach is a preferred alternative over the DFT in cases where spectral analysis of sampled signals is required.