Constrained Hardy space approximation

We consider the problem of minimizing the distance ‖f−ϕ‖Lp(K)‖f−ϕ‖Lp(K), where KK is a subset of the complex unit circle ∂D∂D and ϕ∈C(K)ϕ∈C(K), subject to the constraint that ff lies in the Hardy space Hp(D)Hp(D) and |f|≤g|f|≤g for some positive function gg. This problem occurs in the context of filter design for causal LTI systems. We show that the optimization problem has a unique solution, which satisfies an extremal property similar to that for the Nehari problem. Moreover, we prove that the minimum of the optimization problem can be approximated by smooth functions. This makes the problem accessible for numerical solution, with which we deal in a follow-up paper.