^{1} Department of Mathematics, Technical University of Denmark^{2} Department of Applied Mathematics and Computer Science, Technical University of Denmark^{3} unknown

Abstract:

A Gabor system is a set of time-frequency shifts$S(g,\Lambda) = \{e^{2\pi i b x} g(x-a)\}_{(a,b) \in \Lambda}$of a function $g \in L^2({\bold R}^d)$.We prove that if a finite union of Gabor systems$\bigcup_{k=1}^r S(g_k,\Lambda_k)$, with arbitrary sequences $\Lambda_k$,forms a frame for $L^2({\bold R}^d)$ then the lower and upper Beurlingdensities of $\Lambda = \bigcup_{k=1}^r \Lambda_k$ satisfy$D^-(\Lambda) \ge 1$ and $D^+(\Lambda) <\infty$.This extends recent work of Ramanathan and Steger.As a special case, our results prove the conjecture that no collection$\bigcup_{k=1}^r \{g_k(x-a)\}_{a \in \Gamma_k}$of pure translates can form a frame for $L^2({\bold R}^d)$.