Mutual Invertibility and Search

Tao, Renji

doi:10.1007/978-3-540-78257-5_2

Cited by 3 publications

(4 citation statements)

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“…The results in [11,14] provide criteria and constructions that work for index sets Γ satisfying (1.7) for some neighborhood U , whereas Theorem 1.2 above requires U to be sufficiently small. We mention that even for a molecular frame for L 2 , the extension of the canonical L 2 -frame expansions to Hardy and Lebesgue spaces is non-automatic in general, and that such frames might fail to yield decompositions of L p for 1 < p < ∞, see, e.g., Tao [56] and Tchamitchian [57].…”

Section: Molecular Decompositions the Coorbit Identification Of Aniso...mentioning

confidence: 99%

Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, I

Koppensteiner¹,

Velthoven²,

Voigtlaender³

2021

Preprint

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This paper provides maximal function characterizations of anisotropic Triebel-Lizorkin spaces associated to general expansive matrices for the full range of parameters p ∈ (0, ∞), q ∈ (0, ∞] and α ∈ R. The equivalent norm is defined in terms of the decay of wavelet coefficients, quantified by a Peetre-type space over a one-parameter dilation group. For the Banach space regime p, q ≥ 1, we use this characterization to prove the existence of frames and Riesz sequences of dual molecules for the Triebel-Lizorkin spaces; the atoms are obtained by translations and anisotropic dilations of a single function, where neither the translation nor dilation parameters are required to belong to a discrete subgroup. Explicit criteria for molecules are given in terms of smoothness, decay and moment conditions.

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Section: Molecular Decompositions the Coorbit Identification Of Aniso...mentioning

confidence: 99%

Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, I

Koppensteiner¹,

Velthoven²,

Voigtlaender³

2021

Preprint

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“…In the absence of spectral invariance, the theory of localized frames does not apply. 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].…”

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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“…where N p is defined by (32), and M p is as in (30). When 1 < p < 1.04 we cannot argue directly that the Neumann series for (ST ) −1 converges, because we cannot show M p < 1 in that range.…”

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confidence: 99%

“…Two counterexamples reveal the subtlety of the problem. Tao [32] showed for each p ∈ (1, 2) that there exists a smooth, compactly supported synthesizer ψ that is "good" on L 2 (generating a wavelet frame there) but is "bad" on L p (having non-bijective frame operator). For p ∈ (2, ∞), Tchamitchian and Lemarié [27, pp.…”

Section: Introductionmentioning

confidence: 99%

Wavelet Frame Bijectivity on Lebesgue and Hardy Spaces

Bui

Laugesen

2013

J Fourier Anal Appl

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Dedicated to our friend and mentor Guido Weiss.Abstract. We prove a sufficient condition for frame-type wavelet series in L p , the Hardy space H 1 , and BMO. For example, functions in these spaces are shown to have expansions in terms of the Mexican hat wavelet, thus giving a strong answer to an old question of Meyer.Bijectivity of the wavelet frame operator acting on Hardy space is established with the help of new frequency-domain estimates on the Calderón-Zygmund constants of the frame kernel.

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Mutual Invertibility and Search

Cited by 3 publications

References 0 publications

Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, I

Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, I

On dual molecules and convolution-dominated operators

Wavelet Frame Bijectivity on Lebesgue and Hardy Spaces

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