Gabor windows supported on \([ - 1, 1]\) and dual windows with small support (Q452322)

The Gabor frame is the frame obtained from the translation and modulation of a window function. For the reconstruction of a signal from its samples, the dual frame is needed, which is again the frame of translation and modulation of a window function called the dual window. The smaller the support the better the window for practical applications, but it has some price. For a window function supported on \([-1,1],\) it is known that the compactly supported dual window exists. The authors show that under some extra conditions, the size of the dual window can be reduced by one half. They show that the window of smaller support than previously known exists for the translation parameter \(1\) and the modulation parameter \( b < \frac{2 N}{2N+1}\) for some \( N \in \mathbb N,\) assuming the window function to be a continuous function that has finite number of zeros in \([-1,1]\) such that \(f(x)\neq 0, x\in \left[\frac{N}{b}-N-1,-\frac{N}{b}+N+1\right].\) Under these conditions they show that there exists a continuous dual window supported on [-N, N] if the modulation parameter \(b < \frac{2N}{2N+1}.\) They show that such a window function is optimal in the sense that if \(b=\frac{2N}{2N+1}\) the window function can't be continuous and if \(b > \frac{2N}{2N+1},\) the dual window doesn't exist. The results are illustrated with nice examples.