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 $L^p\\pi (1 < p < \\infty)$ 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 $L^2\\pi$, 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.