Frames of multi-windowed exponentials on subsets of RdRd

Given discrete subsets Λj⊂RdΛj⊂Rd, j=1,…,qj=1,…,q, consider the set of windowed exponentials ⋃j=1q{gj(x)e2πi〈λ,x〉:λ∈Λj} on L2(Ω)L2(Ω). We show that a necessary and sufficient condition for the windows gjgj to form a frame of windowed exponentials for L2(Ω)L2(Ω) with some ΛjΛj is that 0