Frames of multi-windowed exponentials on subsets of ${\mathbb R}^d$

Given discrete subsets $Λ_j\subset {\Bbb R}^d$, $j=1,...,q$, consider the set of windowed exponentials $\bigcup_{j=1}^{q}\{g_j(x)e^{2πi <λ,x>}: λ\inΛ_j\}$ 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 $Λ_j$ is that $m\leq \max_{j\in J}|g_j|\leq M$ 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 $L^2(Ω)$. At the same time, we also construct examples of unbounded sets with finite measure that have no tight exponential frame.

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