Gabor frames with trigonometric spline dual windows

Kim, Inmi

doi:10.1142/s1793557115500722

Cited by 12 publications

(17 citation statements)

References 24 publications

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“…In our pseudo-spline example the order of smoothness can be increased independent of the support size and therefore of the modulation lattice. In [23,27,78,81] the authors construct dual windows that overcome the problem of support in [24]. The paper [81] gives several constructions of Gabor windows using spline functions and discusses the smoothness of the constructed windows and choice of lattices.…”

Section: Related Workmentioning

confidence: 99%

“…However, it involves complicated symbolic computations, especially when the smoothness of the window is increased. In [23,78], the authors are motivated from the solution of a(π ω)b(π ω) +â(π(ω + 1))b(π(ω + 1)) = 1 (4.45) to construct dual windows of equal support size. Several of their windows coincide with ours, which is not surprising since the trigonometric polynomials we construct from (4.42) solve (4.45).…”

Section: Related Workmentioning

confidence: 99%

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Duality for Frames

Fan

Heinecke

Shen

2015

J Fourier Anal Appl

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The subject of this article is the duality principle, which, well beyond its stand at the heart of Gabor analysis, is a universal principle in frame theory that gives insight into many phenomena. Its fiber matrix formulation for Gabor systems is the driving principle behind seemingly different results. We show how the classical duality identities, operator representations and constructions for dual Gabor frames are in fact aspects of the dual Gramian matrix fiberization and its sole duality principle, giving a unified view to all of them. We show that the same duality principle, via dual Gramian matrix analysis, holds for dual (or bi-) systems in abstract Hilbert spaces. The essence of the duality principle is the unitary equivalence of the frame operator and the Gramian of certain adjoint systems. An immediate consequence is, for example, that, even on this level of generality, dual frames are characterized in terms of biorthogonality relations of adjoint systems. We formulate the duality principle for irregular Gabor systems which have no structure whatsoever to the sampling of the shifts and modulations of the generating window. In case the shifts and modulations are sampled from lattices we show how the abstract matrices can be reduced to the simple structured fiber matrices of shift-invariant systems, thus arriving back in the well understood territory. Moreover, in the arena of multiresolution analysis J Fourier Anal Appl (MRA)-wavelet frames, the mixed unitary extension principle can be viewed as the duality principle in a sequence space. This perspective leads to a construction scheme for dual wavelet frames which is strikingly simple in the sense that it only needs the completion of an invertible constant matrix. Under minimal conditions on the MRA, our construction guarantees the existence and easy constructability of non-separable multivariate dual MRA-wavelet frames. The wavelets have compact support and we show examples for multivariate interpolatory refinable functions. Finally, we generalize the duality principle to the case of transforms that are no longer defined by discrete systems, but may have discrete adjoint systems.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Duality for Frames

Fan

Heinecke

Shen

2015

J Fourier Anal Appl

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“…3 we will now construct pairs of dual Gabor frames in L 2 (R d ) with attractive properties. Other constructions in L 2 (R d ) in the literature include [2] and [15]; we will comment on these along the way. Note also the approach (in L 2 (R)) by Laugesen in [18].…”

Section: Construction Of Gabor Frames In L 2 (R D )mentioning

confidence: 99%

“…We avoid a complicated book keeping; and we avoid to enlarge the support of the dual window and can keep the same support size as for the window itself. The construction by I. Kim in [15,16] also provides dual windows with the same support as the given window, but without an explicit expression for the dual window. (iii) Small adjustments of Theorem 4.1 lead to constructions of tight frames.…”

Section: For Some (N Z) D -Periodic Real-valued Trigonometric Polynommentioning

confidence: 99%

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On Partition of Unities Generated by Entire Functions and Gabor Frames in $$ L^2({\mathbb R}^d) $$ L 2 ( R d ) and $$\ell ^2({\mathbb Z}^d)$$ ℓ 2 ( Z d )

Christensen

Kim

2015

J Fourier Anal Appl

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We characterize the entire functions P of d variables, d ≥ 2, for which the Z d -translates of Pχ [0,N ] d satisfy the partition of unity for some N ∈ N. In contrast to the one-dimensional case, these entire functions are not necessarily periodic. In the case where P is a trigonometric polynomial, we characterize the maximal smoothness of Pχ [0,N ] d , as well as the function that achieves it. A number of especially attractive constructions are achieved, e.g., of trigonometric polynomials leading to any desired (finite) regularity for a fixed support size. As an application we obtain easy constructions of matrix-generated Gabor frames in L 2 (R d ), with small support and high smoothness. By sampling this yields dual pairs of finite Gabor frames in 2 (Z d ).

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Spline function generating Gabor systems on the half real line

Yang

2023

Math Methods in App Sciences

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Recently, Gabor analysis on locally compact abelian (LCA) groups has interested some mathematicians. The half real line R + = (0, ∞) is an LCA group under multiplication and the usual topology. This paper addresses spline Gabor frames for L 2 (R + , d𝜇), where 𝜇 is the corresponding Haar measure. We introduce the concept of spline functions on R + by 𝜇-convolution and estimate their Gabor frame sets, that is, lattice sets such that spline generating Gabor systems are frames for L 2 (R + , d𝜇). For an arbitrary spline Gabor frame with special lattices, we present its one dual Gabor frame window, which has the same smoothness as the initial window function. For a class of special spline Gabor Bessel sequences, we prove that they can be extended to a tight Gabor frame by adding a new window function, which has compact support and same smoothness as the initial windows. And we also demonstrate that two spline Gabor Bessel sequences can always be extended to a pair of dual Gabor frames with the adding window functions being compactly supported and having the same smoothness as the initial windows.

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Gabor frames with trigonometric spline dual windows

Cited by 12 publications

References 24 publications

Duality for Frames

Duality for Frames

On Partition of Unities Generated by Entire Functions and Gabor Frames in $$ L^2({\mathbb R}^d) $$ L 2 ( R d ) and $$\ell ^2({\mathbb Z}^d)$$ ℓ 2 ( Z d )

Spline function generating Gabor systems on the half real line

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