From dual pairs of Gabor frames to dual pairs of wavelet frames and vice versa

We discuss an elementary procedure that allows us to construct dual pairs of wavelet frames based on certain dual pairs of Gabor frames and vice versa. The construction preserves tightness of the involved frames. Starting with Gabor frames generated by characteristic functions the construction leads to a class of tight wavelet frames that include the Shannon (orthonormal) wavelet, and applying the construction to Gabor frames generated by certain exponential B-splines yields wavelet frames generated by functions whose Fourier transforms are compactly supported splines with geometrically distributed knot sequences. On the other hand, the pendant of the Meyer wavelet turns out to be a tight Gabor frame generated by a C∞(R)C∞(R) function with compact support. As an application of our results we show that for each given pair of bandlimited dual wavelet frames it is possible to construct dual wavelet frames for any desired scaling and translation parameters.