Wavelet Frame Bijectivity on Lebesgue and Hardy Spaces

Bui, Huy-Qui; Laugesen, Richard S.

doi:10.1007/s00041-013-9268-3

Cited by 15 publications

(15 citation statements)

References 37 publications

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“…Indeed, it is known that the frame operator associated with certain smooth and fast-decaying wavelets with several vanishing moments fails to be invertible on L p -spaces [46,Chapter 4]. In connection to this point, we mention that the Mexican hat wavelet satisfies Daubechies criterion, but the validity of the corresponding L p expansions was established only recently with significant ad-hoc work [15].…”

Section: Related Workmentioning

confidence: 95%

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Invertibility of Frame Operators on Besov-Type Decomposition Spaces

2022

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We derive an extension of the Walnut–Daubechies criterion for the invertibility of frame operators. The criterion concerns general reproducing systems and Besov-type spaces. As an application, we conclude that $$L^2$$ L 2 frame expansions associated with smooth and fast-decaying reproducing systems on sufficiently fine lattices extend to Besov-type spaces. This simplifies and improves recent results on the existence of atomic decompositions, which only provide a particular dual reproducing system with suitable properties. In contrast, we conclude that the $$L^2$$ L 2 canonical frame expansions extend to many other function spaces, and, therefore, operations such as analyzing using the frame, thresholding the resulting coefficients, and then synthesizing using the canonical dual frame are bounded on these spaces.

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Section: Related Workmentioning

confidence: 95%

“…Remark 5. 15 The formulation of Proposition 5.14 is rather technical, because, under those assumptions, the formula defining the frame operator might not make sense for f ∈ D(Q, L 2 , 2 w ). Indeed, the hypothesis are satisfied for every tight frame, even if…”

Section: Proofmentioning

confidence: 99%

Invertibility of Frame Operators on Besov-Type Decomposition Spaces

2022

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“…For example, there are smooth and fast-decaying wavelets with several vanishing moments which yield a frame for L 2 (R), but do not admit a dual frame formed by molecules of a similar quality, and, indeed, do not lead to L p -expansions [89,90]; see also [69,Section 9.2] and [88]. For certain particular wavelet construction schemes, or for specific wavelets, such as the Mexican hat, establishing the validity of L p -expansions is significantly more challenging than that of L 2 expansions [12][13][14][15]. 1.2.3.…”

Section: Introductionmentioning

confidence: 99%

On dual molecules and convolution-dominated operators

Romero¹,

Velthoven²,

Voigtlaender³

2020

Preprint

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We show that sampling or interpolation formulas in reproducing kernel Hilbert spaces can be obtained by reproducing kernels whose dual systems form molecules, ensuring that the size profile of a function is fully reflected by the size profile of its sampled values. The main tool is a local holomorphic calculus for convolution-dominated operators, valid for groups with possibly non-polynomial growth. Applied to the matrix coefficients of a group representation, our methods improve on classical results on atomic decompositions and bridge a gap between abstract and concrete methods.

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“…Thus if we had explicit interpolation bounds on I − T , then we would know how close T must lie to the identity on H 1 and L 2 in order to ensure that the Neumann series for (I − T ) −1 converges on L p . Such questions arose recently in our work on wavelet frame operators [1].…”

Section: Introductionmentioning

confidence: 99%

Explicit Interpolation Bounds Between Hardy space And

Bui¹,

Laugesen

2013

J. Aust. Math. Soc.

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Every bounded linear operator that maps H 1 to L 1 and L 2 to L 2 is bounded from L p to L p for each p ∈ (1, 2), by a famous interpolation result of Fefferman and Stein. We prove L p -norm bounds that grow like O(1/(p − 1)) as p ↓ 1. This growth rate is optimal, and improves significantly on the previously known exponential bound O(2 1/(p−1) ). For p ∈ (2, ∞), we prove explicit L p estimates on each bounded linear operator mapping L ∞ to bounded mean oscillation (BMO) and L 2 to L 2 . This BMO interpolation result implies the H 1 result above, by duality. In addition, we obtain stronger results by working with dyadic H 1 and dyadic BMO. The proofs proceed by complex interpolation, after we develop an optimal dyadic 'good lambda' inequality for the dyadic -maximal operator.2010 Mathematics subject classification: primary 42B30; secondary 46B70.

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Wavelet Frame Bijectivity on Lebesgue and Hardy Spaces

Cited by 15 publications

References 37 publications

Invertibility of Frame Operators on Besov-Type Decomposition Spaces

Invertibility of Frame Operators on Besov-Type Decomposition Spaces

On dual molecules and convolution-dominated operators

Explicit Interpolation Bounds Between Hardy space And

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