This theorem was originally stated independently by Balian [6] and Low [23] for orthogonal systems, but both of their proofs contained a gap, which was later filled by Coifman et al. [11] who also generalized it to Riesz bases. For general references on the Balian–Low Theorem we refer the reader to [8,19]. In [8], the authors also state and prove the so called Amalgam Balian–Low Theorem, which states that if (ϕ, αZ×βZ) is a Riesz basis for L2(R), then ϕ cannot belong to the Feichtinger algebra S0(R), a class of functions decaying well in time and frequency. For a definition of S0(R) see (2) below. Note that the Amalgam Balian–Low Theorem is seemingly weaker than the Balian–Low Theorem, but is not implied by it.
