Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt's ($A_p$) condition

Lyubarskii, Yurii; Seip, Kristian

doi:10.4171/rmi/224

Cited by 55 publications

(55 citation statements)

References 5 publications

(8 reference statements)

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“…For the case of complete interpolating sequences in P W p τ , i.e., interpolating sequences for which the interpolating functions are unique, these are characterized in [15] appealing to the Carleson condition and the Muckenhoupt (A p )-condition for some function associated with the generating function of S. Sufficient conditions are pointed out in [24] using a kind of uniform non-uniqueness condition in the spirit of Beurling. Such a condition cannot be necessary since there are complete interpolating sequences in the Paley-Wiener spaces (which are in particular uniqueness sets).…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

“…The following result, which we will prove below, is nothing but a re-interpretation of [15]. So, in the scale of Paley-Wiener spaces -which represents a subclass of backward shift invariant subspaces -an interpolating sequence is not necessarily interpolating in an arbitrary bigger space, and so a fortiori a dual bounded sequence for a given p is not necessarily interpolating for a bigger space K s I , s < p. This should motivate why in our main result discussed in the next section we increase the space in two directions to get interpolation from dual boundedness: we increase the space by multiplying factors to the defining inner function and by decreasing p.…”

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confidence: 99%

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Uniform minimality, unconditionality and interpolation in backward shift invariant subspaces

Amar

Hartmann

2010

Ann. inst. Fourier

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We discuss relations between uniform minimality, unconditionality and interpolation for families of reproducing kernels in backward shift invariant subspaces. This class of spaces contains as prominent examples the Paley-Wiener spaces for which it is known that uniform minimality does in general neither imply interpolation nor unconditionality. Hence, contrarily to the situation of standard Hardy spaces (and other scales of spaces), changing the size of the space seems in this context necessary to deduce unconditionality or interpolation from uniform minimality. Such a change can take two directions: lowering the power of integration, or "increasing" the defining inner function (e.g. increasing the type in the case of Paley-Wiener space).

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Section: Annales De L'institut Fouriermentioning

confidence: 99%

mentioning

confidence: 99%

Uniform minimality, unconditionality and interpolation in backward shift invariant subspaces

Amar

Hartmann

2010

Ann. inst. Fourier

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“…Our next theorem generalizes the main theorem of [12]. We should add that the problem of describing complete interpolating sequences for L 2 τ | z| was investigated for the first time in [16].…”

Section: ν(I) µ(I)mentioning

confidence: 97%

“…We should add that the problem of describing complete interpolating sequences for L 2 τ | z| was investigated for the first time in [16]. We refer to [12] and [8] for the extensive history of the subject.…”

Section: ν(I) µ(I)mentioning

confidence: 99%

“…This proof makes use of an interpolation theorem of de Branges for the space H(E), as well as a theorem from Section 6 describing socalled complete interpolating sequences for weighted Paley-Wiener spaces in terms of Muckenhoupt's (A 2 ) condition (cf. [12]). It is the latter relation which explains the appearance of the (A 2 ) condition in Theorem 4.…”

Section: T)) Z/t]m(t)dtmentioning

confidence: 99%

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Weighted Paley-Wiener spaces

Lyubarskii

Seip

2002

J. Amer. Math. Soc.

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We study problems of sampling and interpolation in a wide class of weighted spaces of entire functions. These weights are characterized by the property that their natural regularization as the envelop of the unit ball of the corresponding space is equivalent to the original weight. We give an independent description of such weights and also show that, in a sense, this is the widest class of weights and associated spaces for which results on sets of uniqueness, sampling, and interpolation related to the classical Paley-Wiener spaces can be extended in a direct and natural way, keeping the basic features of the theory intact. One of the basic tools for our study is the De Brange theory of spaces of entire functions.

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A Direct Interpolation Method for Irregular Sampling

Flornes

Lyubarskii

Seip

1999

Applied and Computational Harmonic Analysis

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In this paper, band-limited functions are reconstructed from their values taken at a sequence of irregularly spaced sample points. We use a modified Lagrange formula, which is attributed to Boas and Bernstein. The formula used in this paper differs from the classical Boas-Bernstein formula in the following way. Instead of using infinite canonical products with respect to the whole sequence of sample points, we use canonical products with respect to sequences of sample points which are irregularly spaced only on finite intervals. Estimates for the truncation error of this reconstruction method are given.

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Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt's ($A_p$) condition

Cited by 55 publications

References 5 publications

Uniform minimality, unconditionality and interpolation in backward shift invariant subspaces

Uniform minimality, unconditionality and interpolation in backward shift invariant subspaces

Weighted Paley-Wiener spaces

A Direct Interpolation Method for Irregular Sampling

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