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

Abstract

We describe the complete interpolating sequences for the Paley-Wiener spaces $L_{\pi }^p(1<p<\infty )$ in terms of Muckenhoupt's ($A_p$) condition. For $p=2$, this description coincides with those given by Pavlov [9], Nikol'skii [8], and Minkin [7] of the unconditional bases of complex exponentials in $L^2(–\pi ,\pi )$. While the techniques of these authors are linked to the Hilbert space geometry of $L_{\pi }^2$, our method of proof is based on turning the problem into one about boundedness of the Hilbert transform in certain weighted $L^p$ spaces of functions and sequences.