  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 (1979), Nikol'skii
(1980), and Minkin (1992) 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.
