Fitting an Escalier to a Curve

We analyze the problem of fitting a fonction en escalier or multi-step function to a curve in L^2 Hilbert space. We propose a two-stage optimization approach whereby the step positions are initially fixed, corresponding to a classic linear least-squares problem with closed-form solution, and then are allowed to vary, leading to first-order conditions that can be solved recursively. We find that, subject to regularity conditions, the speed of convergence is linear as the number of steps $n$ goes to infinity, and we develop a simple algorithm to recover the global optimum fit. Our numerical results based on a sweep search implementation show promising performance in terms of speed and accuracy.