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On Gabor frames generated by sign-changing windows and B-splines
Christensen, O.,Kim, H.O.,Kim, R.Y. Academic Press 2015 Applied and computational harmonic analysis Vol.39 No.3
For a class of compactly supported windows we characterize the frame property for a Gabor system {E<SUB>mb</SUB>T<SUB>na</SUB>g}<SUB>m,n@?Z</SUB>, for translation parameters a belonging to a certain range depending on the support size. We show that the obstructions to the frame property are located on a countable number of ''curves.'' For functions that are positive on the interior of the support these obstructions do not appear, and the considered region in the (a,b) plane is fully contained in the frame set. In particular this confirms a recent conjecture about B-splines by Grochenig in that particular region. We prove that the full conjecture is true if it can be proved in a certain ''hyperbolic strip.''
TIGHT MATRIX-GENERATED GABOR FRAMES IN $L^2(\mathbb{R}^d)$ WITH DESIRED TIME-FREQUENCY LOCALIZATION
Christensen, Ole,Kim, Rae-Young Korean Society of Computational and Applied Mathem 2008 Journal of applied mathematics & informatics Vol.26 No.5
Based on two real and invertible $d{\times}d$ matrices Band C such that the norm $||C^T\;B||$ is sufficiently small, we provide a construction of tight Gabor frames $\{E_{Bm}T_{Cn}g\}_{m,n{\in}{\mathbb{Z}^d}$ with explicitly given and compactly supported generators. The generators can be chosen with arbitrary polynomial decay in the frequency domain.
Extensions of Bessel sequences to dual pairs of frames
Christensen, O.,Kim, H.O.,Kim, R.Y. Academic Press 2013 Applied and computational harmonic analysis Vol.34 No.2
Tight frames in Hilbert spaces have been studied intensively for the past years. In this paper we demonstrate that it often is an advantage to use pairs of dual frames rather than tight frames. We show that in any separable Hilbert space, any pairs of Bessel sequences can be extended to a pair of dual frames. If the given Bessel sequences are Gabor systems in L<SUP>2</SUP>®, the extension can be chosen to have Gabor structure as well. We also show that if the generators of the given Gabor Bessel sequences are compactly supported, we can choose the generators of the added Gabor systems to be compactly supported as well. This is a significant improvement compared to the extension of a Bessel sequence to a tight frame, where the added generator only can be compactly supported in some special cases. We also analyze the wavelet case, and find sufficient conditions under which a pair of wavelet systems can be extended to a pair of dual frames.