A Fredholm integral operator for the differentiation problem

We propose a Fredholm integral equation of the first kind to reformulate the differentiation problem. For the corresponding integral operator, we present an equation that characterizes the eigenvalues and eigenfunctions for each derivation order. For the first- and second-order differentiation an explicit formulation of the eigensystem is provided. The results of this spectral analysis are used to devise an algorithm for the numerical differentiation. We also report the results of a numerical experiment with exact data and with noise-contaminated data to show the convergence and the stability properties of the proposed algorithm.