Consider a continuous function g ∈ L 2(ℝ) that is supported on [ − 1, 1] and generates a Gabor frame with translation parameter 1 and modulation parameter 0 for some N ∈ ℕ. Under an extra condition on the zeroset of the window g we show that there exists a continuous dual window supported on [ − N, N]. We also show that this result is optimal: indeed, if b2N2N+1 then a dual window supported on [ − N, N] does not exist. In the limit case b=2N2N+1 a dual window supported on [ − N, N] might exist, but cannot be continuous.
- Gabor frame
- Compactly supported window
- Compactly supported dual window
Christensen, O., Kim, H. O., & Kim, R. Y. (2012). Gabor windows supported on [ − 1, 1] and dual windows with small support. Advances in Computational Mathematics, 36(4), 525-545. https://doi.org/10.1007/s10444-011-9189-0