Constrained Hardy Space Approximation II: Numerics

In a previous paper [Constrained Hardy Space Approximation, preprint available at http://www.mathematik.uni-karlsruhe.de/iwrmm/seite/ preprints/] we considered the problem of minimizing the distance ‖f − φ‖Lp(K), where K is a subset of the complex unit circle ∂D and φ ∈ C(K), subject to the constraint that f lies in the Hardy space H(D) and |f | ≤ g for some positive function g. This problem occurs in the context of filter design for causal LTI systems. In this paper we devise a general discretization scheme for this problem and show convergence as the discretization becomes better. We derive several concrete discretizations and cast them in the form of second-order cone programs, which can be solved efficiently. We demonstrate this practically with a problem from the design of dispersion compensating mirrors for the generation of ultrashort laser pulses. A MATLAB implementation of our method is available at http://www.mathematik.uni-karlsruhe.de/grk1294/~schneck/.