摘要 :
Structured frames in L-2(R), such as wavelet and Gabor frames, have been extensively studied. But the analysis on L-2(R+) with R+ = (0, infinity) has been rarely reported. It is because R is a group under addition but R+ is not. T...
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Structured frames in L-2(R), such as wavelet and Gabor frames, have been extensively studied. But the analysis on L-2(R+) with R+ = (0, infinity) has been rarely reported. It is because R is a group under addition but R+ is not. This results in L-2(R+) admitting no traditional wavelet or Gabor frames. Recently, the concept of F-a-frame in L-2(R+) was introduced and studied. In this paper, we present some identities and inequalities for F-a-frame sequences in L-2(R+).
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摘要 :
Structured frames in L-2(R), such as wavelet and Gabor frames, have been extensively studied. But the analysis on L-2(R+) with R+ = (0, infinity) has been rarely reported. It is because R is a group under addition but R+ is not. T...
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Structured frames in L-2(R), such as wavelet and Gabor frames, have been extensively studied. But the analysis on L-2(R+) with R+ = (0, infinity) has been rarely reported. It is because R is a group under addition but R+ is not. This results in L-2(R+) admitting no traditional wavelet or Gabor frames. Recently, the concept of F-a-frame in L-2(R+) was introduced and studied. In this paper, we present some identities and inequalities for F-a-frame sequences in L-2(R+).
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Casazza, Han and Larson characterized various properties of the direct sum of two frame sequences. We add characterizations of other properties and study the relationship between the direct sum and the sum of frame sequences. In p...
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Casazza, Han and Larson characterized various properties of the direct sum of two frame sequences. We add characterizations of other properties and study the relationship between the direct sum and the sum of frame sequences. In particular, we find a necessary and sufficient condition for the sum of two strongly disjoint (orthogonal) frame sequences (in the same Hilbert space) to be a frame sequence, and thereby show that the sum of two strongly disjoint frame sequences may not be a frame sequence. We also show that the closedness of the sum of the synthesis operators of two frame sequences and that of the sum of the frame operators of the same frame sequences are not related. Other observations are also included.
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Casazza and Kutyniok [Frames of subspaces, in Wavelets, Frames and Operator Theory, Contemporary Mathematics, Vol. 345 (American Mathematical Society, Providence, RI, 2004), pp. 87-113] defined fusion frames in Hilbert spaces to s...
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Casazza and Kutyniok [Frames of subspaces, in Wavelets, Frames and Operator Theory, Contemporary Mathematics, Vol. 345 (American Mathematical Society, Providence, RI, 2004), pp. 87-113] defined fusion frames in Hilbert spaces to split a large frame system into a set of (overlapping) much smaller systems and being able to process the data effectively locally within each sub-system. In this paper, we handle this problem using block sequences and generalized block sequences with respect to g-frames. Examples have been given to show their existence. A necessary and sufficient condition for a block sequence with respect to a g-frame to be a g-frame has been given. Finally, a sufficient condition for a generalized block sequence with respect to a g-frame to be a g-frame has been given.
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We present approximations of a large class of Weyl-Heisenberg frame operators on by Gabor Bessel sequence operators generated by compactly supported smooth functions.
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In this paper, we first introduce the notion of controlled weaving K-g-frames in Hilbert spaces. Then, we present sufficient conditions for controlled weaving K-g-frames in separable Hilbert spaces. Also, a characterization of con...
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In this paper, we first introduce the notion of controlled weaving K-g-frames in Hilbert spaces. Then, we present sufficient conditions for controlled weaving K-g-frames in separable Hilbert spaces. Also, a characterization of controlled weaving K-g-frames is given in terms of an operator. Finally, we show that if bounds of frames associated with atomic spaces are positively confined, then controlled K-g-woven frames give ordinary weaving K-frames and vice-versa.
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We show that, if A and B are bounded operators on a Hilbert space and X and Y are strongly disjoint (orthogonal) frame sequences, then A(X) + B(Y) is a frame sequence if and only if the sum of the ranges of the synthesis operators...
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We show that, if A and B are bounded operators on a Hilbert space and X and Y are strongly disjoint (orthogonal) frame sequences, then A(X) + B(Y) is a frame sequence if and only if the sum of the ranges of the synthesis operators of A( X) and B(Y) is closed. We also show that, given two disjoint frame sequences, the sum is a frame sequence if the sum of the ranges of the synthesis operators is closed but not vice versa. A counterexample is given by a couple of frames of shifts for two finitely generated shift-invariant spaces of L-2(R).
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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 Hilber...
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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~2(R), 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.
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Frame and fusion frame high-resolution image fusion formulations are presented. These techniques use the physical point spread function (PSF) of cameras as the building block of the mathematical frames in the fusion process. Camer...
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Frame and fusion frame high-resolution image fusion formulations are presented. These techniques use the physical point spread function (PSF) of cameras as the building block of the mathematical frames in the fusion process. Cameras producing the low-resolution images are allowed to be different, and thereby possess different PSFs. Fused image reconstructions are carried out by a dimension invariance principle and by a set of iterative reconstruction algorithms. These frame fundamental approaches are also seen to be robust to realistic fusion problems from inhomogeneous image measurements (taken at different space or time or by different cameras), which is one of the main focuses of this paper. The effectiveness of this approach is demonstrated through both simulated and realistic examples. The results are quite encouraging.
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From the perspectives of duality and extensions, Gabor frames and wavelet frames have contrasting behaviour. Our chief concern here is about duality. Canonical duals of wavelet frames may not be wavelet frames, whereas canonical d...
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From the perspectives of duality and extensions, Gabor frames and wavelet frames have contrasting behaviour. Our chief concern here is about duality. Canonical duals of wavelet frames may not be wavelet frames, whereas canonical duals of Gabor frames are Gabor frames. Keeping these in view, we give several constructions of wavelet frames with wavelet canonical duals. For this, a simple characterisation of Bessel sequences and a general commutativity result are given, the former also leading naturally to some extension results. (C) 2018 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
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