Frames , Wavelets , and Tomography

For Gabor sets, (g; a, b), it is known that (g; a, b) is a frame if and only if (g; 1/b, 1/a) is a Riesz basis for its span. In particular, for every g there is a0 such that for every a < a0, there is a bm = bm(a) > 0 so that for every b < bm, (g; a, b) is a frame, and (g; 1/b, 1/a) is a Riesz basis sequence. In this talk we shall consider a similar problem for wavelet sets (Ψ; a, b). The main result reads as follows. Given an admissible Ψ, there is a0 > 0 so that for every a < a0, there is a bm = bm(a) so that for every b < bm, (Ψ; a, b) is a frame, and (Ψ; 1/b, 1/a) is a Riesz basis for its span. (3) Boncek, John ([email protected]) Title: Generalization of Frame Potentials Abstract # : 984-43-222 Abstract: The concept of frame potentials has recently emerged as a useful technique for The concept of frame potentials has recently emerged as a useful technique for discovering and analyzing tight frames. In our work, we introduce variations on this definition and study the corresponding frame behaviors. (4) Bownik, Marcin Title: Affine Frames, GMRA’s, and the Canonical Dual. Joint with Eric Weber Abstract # : 984-42-152 Abstract: We show that if the canonical dual of an affine frame has the affine structure, with the same number of generators, then the core subspace V0 is shift invariant. We demonstrate, however, that the converse is not true. We apply our results in the setting of oversampling affine frames, as well as computing the period of a Riesz wavelet, which answers in the affir: 984-42-152 Abstract: We show that if the canonical dual of an affine frame has the affine structure, with the same number of generators, then the core subspace V0 is shift invariant. We demonstrate, however, that the converse is not true. We apply our results in the setting of oversampling affine frames, as well as computing the period of a Riesz wavelet, which answers in the affirmative a conjecture of Daubechies and Han. Additionally, we completely characterize when the canonical dual of a quasi-affine frame has the quasi-affine structure.