Given discrete subsets Λj⊂Rd, j=1,...,q, consider
the set of windowed exponentials ⋃j=1q{gj(x)e2πi:λ∈Λj} on L2(Ω). We show that a necessary
and sufficient condition for the windows gj to form a frame of windowed
exponentials for L2(Ω) with some Λj is that m≤maxj∈J∣gj∣≤M almost everywhere on Ω for some subset J of {1,...,q}. If Ω is unbounded, we show that there is no frame of windowed
exponentials if the Lebesgue measure of Ω is infinite. If Ω is
unbounded but of finite measure, we give a sufficient condition for the
existence of Fourier frames on L2(Ω). At the same time, we also
construct examples of unbounded sets with finite measure that have no tight
exponential frame