Gabor windows supported on [-1,1] and compactly supported dual windows

Consider a bounded function g supported on [-1, 1] and a modulation parameter b is an element of inverted left perpendicular1/2, 1inverted right perpendicular for which the Gabor system {E(mb)T(n)g}(m,n is an element of Z) is a frame. We show that such a frame always has a compactly supported dual window. More precisely, we show that if b <N/N+1 for some N is an element of N, it is possible to find a dual window supported on [-N, N]. Under the additional assumption that g is continuous and only has a finite number of zeros on inverted left perpendicular-1, 1inverted right perpendicular, we characterize the frame property of {E(mb)T(n)g}(m,n is an element of Z). As a consequence we obtain easily verifiable criteria for a function g to generate a Gabor frame with a dual window having compact support of prescribed size.