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Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt’s \((A_p)\) condition. (English) Zbl 0918.42003
The Paley-Wiener spaces \(L^{p}_{\pi}\) \((1<p<\infty)\) consist of all entire functions of exponential type at most \(\pi\) whose restrictions to \(\mathbb R\) are in \(L^{p}\). The authors study complete interpolating sequences for \(L^{p}_{\pi}\), that is those sequences \(\Lambda =\{\lambda_{k}\}, \lambda_{k} =\xi _{k} + i\eta _{k}\) in \( \mathbb C\) for which the problem \(f(\lambda _{k}) = a_{k}\) has a unique solution \(f\in L^{p}_{\pi}\) for every sequence \(\{a_{k}\}\) satisfying \(\sum _{k } | a_{k}| ^{p}e^{-p\pi | \eta _{k}| } (1+ | \eta _{k}|) < \infty\). In the case \(p=2\), this problem is equivalent to that of describing all unconditional bases in \(L^{2}(-\pi , \pi)\) of the form \(\{\exp (i\lambda _{k}t)\}\). The \(p=2\) case was solved in various stages by Pavlov, Nikol’skij and Minkin. Each author made critical use of the Hilbert space geometry of \(L^{2}_{\pi}\). In this paper the authors’ method is dependent on the boundedness of the Hilbert transform in certain weighted \(L^{p}\) spaces of functions and sequences. Their proof holds for all \(p\), \(1<p < \infty\) and also shows that for \(p= \infty\) or \(0<p\leq 1\) there are no complete interpolating sequences.

MSC:
42A15 Trigonometric interpolation
41A05 Interpolation in approximation theory
44A15 Special integral transforms (Legendre, Hilbert, etc.)
30E05 Moment problems and interpolation problems in the complex plane
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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