Sparse approximation of functions using sums of exponentials and AAK theory

We consider the problem of approximating functions by sums of few exponentials functions, either on an interval or on the positive half-axis. We study both continuous and discrete cases, i.e. when the function is replaced by a number of equidistant samples. Recently, an algorithm has been constructed by Beylkin and Monzón for the discrete case. We provide a theoretical framework for understanding how this algorithm relates to the continuous case.