CHARACTERIZATIONS OF GABOR FRAMES

For positive integers N, M ∈ N, let g n , ∈ L²(R) with supp(D b g n ) ∪ supp(D b g n ) ⊂ [0, M], where D b g(t) := b ?1/2 g(t/b), b ＞ 0 and n = 1, 2, ...,N. We give another set of necessary and sufficient conditions for the multiwindow Gabor system (G N ; a, b) := {g n;k,l (t) := g n (t ? ka)e 2πilbt } n=1,...,N; k,l∈Z and the corresponding Gabor system (G N ; a, b) to form a pair of dual frames for the rational sampling ab = P/Q with P,Q ∈ N, in addition to the Zibulski-Zeevi and Janssen conditions. The conditions come from the back transform of Zibulski-Zeevi condition to the primal domain but are more informative for the applications. As an applications, we show that the Gabor system (G N ; 1, 1) forms an orthonormal basis for L²(R) if and only if N = 1 and |g₁(t)| = ∑ M?1 m=0 x m+Em (t) a.e. where {E m m=0,...,M?1 forms Lebesgue measurable partition of the unit interval [0, 1). Our criterions provides a rich family of multiwindow dual Gabor frames and multiwindow tight Gabor frames for L²(R) for the particular choices of P, Q, N, M ∈ N.