Sampling expansions for multiband signals

The class BIof 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 BIis uniquely determined from its uniform samples taken at the rate 1/T samples/s. 2) The set of complex exponentials is complete on the set I. 3) the translated sets with σ = 2π/T do not intersect I for any . 4) Each signal in BIpossesses 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 BI.