Constrained Hardy space approximation

Constrained Hardy space approximation

We consider 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. We show that the optimization problem has a unique solution, which satisfies an extrema...

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 oc...

Constrained extremal problems in the Hardy space H 2 and

We study some approximation problems on a strict subset of the circle by analytic functions of the Hardy space H of the unit disk (in C), whose modulus satisfy a pointwise constraint on the complentary part of the circle. Existence and uniqueness results, as well as pointwise saturation of the constraint, are established. We also derive a critical point equation which gives rise to a dual formu...

Approximation and Spanning in the Hardy Space, by Affine Systems

We find weak conditions on ψ ∈ L(R) with ψ̂(0) = 1 such that every function in the Hardy space is a linear combination of translates and dilates of ψ. More precisely, we prove for each f ∈ H(R) the scale averaged approximation formula f(x) = lim J→∞ 1 J J ∑

On a Rational Approximation Problemin the Real Hardy Space