Sampling expansions for multiband signals

The class B
I
of finite energy signals bandlimited to a set I is considered where I denotes a finite union of disjoint open intervals (bands) on the real frequency axis. If such a signal is sampled uniformly (constant intersample spacing T), it is shown that the following five propositions are equivalent. 1) Any signal in B
I
is uniquely determined from its uniform samples taken at the rate 1/T samples/s. 2) The set of complex exponentials
{e^{in\omegaT}}\min{-infin}\max{\infin}
is complete on the set I. 3) the translated sets
{\omega + k\sigma: \omega\epsilonI}
with σ = 2π/T do not intersect I for any
k \neq 0
. 4) Each signal in B
I
possesses a mean-square convergent sampling expansion in terms of its samples at rate 1/T samples/s. 5) A generalized Parseval relation holds for each pair of signals in B
I
.