Analytic approximation inLpand coinvariant subspaces of the Hardy space

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...

Hardy space of operator-valued analytic functions

We are concerned with Hardy and BMO spaces of operator-valued functions analytic in the unit disk of C. In the case of the Hardy space, we involve the atomic decomposition since the usual argument in the scalar setting is not suitable. Several properties (the Garsia-norm equivalent theorem, Carleson measure, and so on) of BMOA spaces are extended to the operator-valued setting. Then, the operat...

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...

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