Constrained Hardy Space Approximation

We consider the problem of minimizing the distance kf − kLp(K), where K is a subset of the complex unit circle @D and ∈ C(K), subject
to the constraint that f lies in the Hardy space Hp(D) and |f| ≤ g for
some positive function g. 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.