Uniform minimality, unconditionality and interpolation in backward shift invariant subspaces

Amar, Éric; Hartmann, Andreas

doi:10.5802/aif.2575

Cited by 7 publications

(6 citation statements)

References 24 publications

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“…In this generality, however, even the case of

K_{θ}^{p}

with

1 < p < \infty

(or with

p = 2

) cannot be viewed as completely understood. At least, the existing results (see for some of these) appear to be far less clear‐cut than in the current setting. By contrast, the difficulty we have to face here is entirely due to the endpoint position of

K_{B}^{\infty}

within the

K_{B}^{p}

scale, the only enemy being the failure of the Riesz projection theorem.…”

Section: Introduction and Resultscontrasting

confidence: 73%

“…While we are only concerned with the traces of functions from K ∞ B on {a j }, which is the zero sequence of B, an obvious generalization would consist in restricting our functions (or those from K ∞ θ , with θ inner) to an arbitrary interpolating sequence in D. In this generality, however, even the case of K p θ with 1 < p < ∞ (or with p = 2) cannot be viewed as completely understood. At least, the existing results (see [1,6,13] for some of these) appear to be far less clear-cut than in the current setting. By contrast, the difficulty we have to face here is entirely due to the endpoint position of K ∞ B within the K p B scale, the only enemy being the failure of the M. Riesz projection theorem.…”

Section: Introduction and Resultscontrasting

confidence: 65%

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A free interpolation problem for a subspace of H∞

Dyakonov

2018

Bull. London Math. Soc.

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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 .

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“…In this generality, however, even the case of

K_{θ}^{p}

with

1 < p < \infty

(or with

p = 2

K_{B}^{\infty}

within the

K_{B}^{p}

scale, the only enemy being the failure of the Riesz projection theorem.…”

Section: Introduction and Resultscontrasting

confidence: 73%

Section: Introduction and Resultscontrasting

confidence: 65%

A free interpolation problem for a subspace of H∞

Dyakonov

2018

Bull. London Math. Soc.

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“…We shall be concerned with interpolation problems for functions in star-invariant subspaces-specifically, for those in K p B , where B is an interpolating Blaschke product. Some of the earlier results in this area can be found in [1,7,16,18], while others, more relevant to our current topic, will be recalled presently.…”

Section: Introduction and Resultsmentioning

confidence: 83%

“…To describe the trace class K ∞ B | Z in the general case (i.e., when (1.7) no longer holds), we first introduce a bit of notation. Once the interpolating sequence Z = {z j } is fixed, we associate with each sequence W = {w j } from 1 1 (Z) the conjugate sequence W = { w k } whose elements are…”

Section: Introduction and Resultsmentioning

confidence: 99%

Interpolation and duality in spaces of pseudocontinuable functions

Dyakonov

2022

Math. Z.

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Given an inner function $$\theta $$ θ on the unit disk, let $$K^p_\theta :=H^p\cap \theta {\overline{z}}\overline{H^p}$$ K θ p : = H p ∩ θ z ¯ H p ¯ be the associated star-invariant subspace of the Hardy space $$H^p$$ H p . Also, we put $$K_{*\theta }:=K^2_\theta \cap \mathrm{BMO}$$ K ∗ θ : = K θ 2 ∩ BMO . Assuming that $$B=B_{{\mathcal {Z}}}$$ B = B Z is an interpolating Blaschke product with zeros $${\mathcal {Z}}=\{z_j\}$$ Z = { z j } , we characterize, for a number of smoothness classes X, the sequences of values $${\mathcal {W}}=\{w_j\}$$ W = { w j } such that the interpolation problem $$f\big |_{{\mathcal {Z}}}={\mathcal {W}}$$ f | Z = W has a solution f in $$K^2_B\cap X$$ K B 2 ∩ X . Turning to the case of a general inner function $$\theta $$ θ , we further establish a non-duality relation between $$K^1_\theta $$ K θ 1 and $$K_{*\theta }$$ K ∗ θ . Namely, we prove that the latter space is properly contained in the dual of the former, unless $$\theta $$ θ is a finite Blaschke product. From this we derive an amusing non-interpolation result for functions in $$K_{*B}$$ K ∗ B , with $$B=B_{{\mathcal {Z}}}$$ B = B Z as above.

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“…In [AH10], the authors show that uniform minimality implies unconditionality in a bigger space for certain backward shift invariant spaces K p I := H p ∩ IH p 0 (considered here on the unit circle T) for which the Paley-Wiener spaces are a particular case. We will use here a different approach to obtain a stronger result.…”

Section: Introductionmentioning

confidence: 99%

Minimality, (Weighted) Interpolation in Paley-Wiener Spaces and Control Theory

Gaunard

2011

Preprint

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It is well known from a result by Shapiro-Shields that in the Hardy spaces, a sequence of reproducing kernels is uniformly minimal if and only if it is an unconditional basis in its span. This property which can be reformulated in terms of interpolation and so-called weak interpolation is not true in Paley-Wiener spaces in general. Here we show that the Carleson condition on a sequence Λ together with minimality in Paley-Wiener spaces P W p τ of the associated sequence of reproducing kernels implies the interpolation property of Λ in P W p τ +ǫ , for every ǫ > 0. With the same technics, using a result of McPhail, we prove a similary result about minimlity and weighted interpolation in P W p τ +ǫ .. We apply the results to control theory, establishing that, under some hypotheses, a certain weak type of controllability in time τ > 0 implies exact controllability in time τ + ǫ, for every ǫ > 0.

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Uniform minimality, unconditionality and interpolation in backward shift invariant subspaces

Cited by 7 publications

References 24 publications

A free interpolation problem for a subspace of H∞

A free interpolation problem for a subspace of H∞

Interpolation and duality in spaces of pseudocontinuable functions

Minimality, (Weighted) Interpolation in Paley-Wiener Spaces and Control Theory

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