A free interpolation problem for a subspace of H∞

Abstract

Given an inner function θ, the associated star-invariant subspace K ∞ θ is formed by the functions f ∈ H ∞ that annihilate (with respect to the usual pairing) the shift-invariant subspace θH 1 of the Hardy space H 1 . Assuming that B is an interpolating Blaschke product with zeros {aj}, we characterise the traces of functions from K ∞ B on the sequence {aj}. The trace space that arises is, in general, nonideal (that is, the sequences {wj} belonging to it admit no nice description in terms of the size of |wj|), but we do point out explicit -and sharp -size conditions on |wj| which make it possible to solve the interpolation problem f (aj) = wj (j = 1, 2, . . . ) with a function f ∈ K ∞ B .

Interpolation and duality in spaces of pseudocontinuable functions

“…It was further conjectured in [13,14] that the trace space K 1 B Z is describable in similar terms, i.e., that the necessary conditions W ∈ ℓ 1 1 (Z) and W ∈ ℓ 1 1 (Z) are also sufficient for W to be in K 1 B Z . To the best of our knowledge, the conjecture is still open.…”

Interpolation in model spaces

“…In what follows, we will consider two interpolating sequences (a j ) and (z j ) that are ρ-separated or far from each other; that is, with the property that there exists ε > 0 with (5) inf…”

“…We are particularly interested in Blaschke products for which the zero sequence (a j ) is an interpolating sequence for H ∞ . In [5], Dyakonov proved the following: Theorem 1.1 ([5]). Suppose that (α j ) is an ∞ sequence and B is an interpolating Blaschke product with zeros (a j ).…”

Interpolation in Model Spaces

“…Theorem 1.1 ( [5]). Suppose that (α j ) is an ℓ ∞ sequence and B is an interpolating Blaschke product with zeros (a j ).…”