Frames of multi-windowed exponentials on subsets of R~d

摘要

Given discrete subsets ∧_j (∩) R~d, j = 1,..., q, consider the set of windowed exponentials ∪_j~q=1{g_j(x)e~(2πi(λ,x)): λ ∈ ∧_j} on L~2(Ω). We show that a necessary and sufficient condition for the windows g_j to form a frame of windowed exponentials for L~2(Ω) with some ∧_j is that 0 ＜ m ≤ max_(j∈J) |g_j| ≤ 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 simple sufficient condition for the existence of Fourier frames on L~2(Ω). At the same time, we also construct examples of unbounded sets with finite measure that have no tight exponential frame.