Pairs of Dual Gabor Frame Generators with Compact Support and Desired Frequency Localization

Let g ∈ L2(R) be a compactly supported function, whose integertranslates {Tkg}k∈Z form a partition of unity. We prove that for certain translationand modulation parameters, such a function g generates a Gabor frame, with a (non-canonical) dual generated by a finite linear combination h of the functions {Tkg}k∈Z; the coefficients in the linear combination are given explicitly. Thus, h has compact support, and the decay in frequency is controlled by the decay of ĝ. In particular, the result allows the construction of dual pairs of Gabor frames, where both generators are given explicitly, have compact support, and decay fast in the Fourier domain. We further relate the construction to wavelet theory. Letting D denote the dilation operator and BN be the N th order B-spline, our results imply that there exist dual Gabor frames with generators of the type g = ∑ ckDTkBN and h = ∑ c̃kDTkBN , where both sums are finite. It is known that for N > 1, such functions can not generate dual wavelet frames {DTkg}j,k∈Z, {D Tkh}j,k∈Z. ∗The author thanks the Erwin Schrödinger Institute in Vienna for hospitality and support during visits in 2005. AMS Math. Subject classification: 42C15, 42C40.