  We solve the problem of Duffin and Schaeffer (1952) of characterizing those
sequences of real frequencies which generate Fourier frames. Equivalently, we
characterize the sampling sequences for the Paley-Wiener space. The key step is
to connect the problem with de Branges' theory of Hilbert spaces of entire
functions. We show that our description of sampling sequences permits us to
obtain a classical inequality of H. Landau as a consequence of Pavlov's
description of Riesz bases of complex exponentials and the John-Nirenberg
theorem. Finally, we discuss how to transform our description into a working
condition by relating it to an approximation problem for subharmonic functions.
By this approach, we determine the critical growth rate of a nondecreasing
function $\psi$ such that the sequence $\{\lamdda_k\}){k\in \Bbb Z}$ defined by
$\lambda_k + \psi(\lambda_k)=k$ is a sampling.
