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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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