Orthonormal bases for α-modulation spaces

Nielsen, Morten

doi:10.1007/bf03191240

Cited by 8 publications

(10 citation statements)

References 19 publications

(23 reference statements)

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“…As a further application of these brushlet bases for α-modulation spaces, Borup and Nielsen derived boundedness results for certain pseudodifferential operators, as briefly discussed above. In [67], Nielsen generalized the results concerning brushlet bases from the one-dimensional case to the case d = 2. Despite their great utility, we remark that brushlet bases are not generated by a single prototype function; furthermore, brushlets are bandlimited and thus cannot be compactly supported.…”

Section: Since We Clearly Havementioning

confidence: 99%

Structured, compactly supported Banach frame decompositions of decomposition spaces

Voigtlaender

2022

Dissertationes Math.

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Section: Since We Clearly Havementioning

confidence: 99%

Structured, compactly supported Banach frame decompositions of decomposition spaces

Voigtlaender

2022

Dissertationes Math.

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“…As a further application of these brushlet bases for α-modulation spaces, Borup and Nielsen derived boundedness results for certain pseudodifferential operators, as briefly discussed above. In [61], Nielsen generalized the results concerning brushlet bases from the one-dimensional case to the case d = 2. Despite their great utility, we remark that brushlet bases are not generated by a single prototype function; furthermore, brushlets are bandlimited and can thus not be compactly supported.…”

Section: Since We Clearly Havementioning

confidence: 99%

Structured, compactly supported Banach frame decompositions of decomposition spaces

Voigtlaender¹

2016

Preprint

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We present a very general framework for the construction of structured, possibly compactly supported Banach frames and atomic decompositions for a given decomposition space. Here, a decomposition space D Q, L p , ℓ q w is defined essentially like a classical Besov space, but the usual dyadic covering is replaced by an (almost) arbitrary covering. Special cases include the class of Besov spaces and (α-)modulation spaces, as well as a large class of wavelet-type coorbit spaces and so-called shearlet smoothness spaces.Assuming that the covering Q is of the regular form, we fix a prototype function γ ∈ L 1 R d and consider the structured generalized shift invariant systemwhere Lx and M ξ denote translation and modulation, respectively. The main contribution of the paper is to provide verifiable conditions on the prototype γ which ensure that Ψ δ forms, respectively, a Banach frame or an atomic decomposition for the space D Q, L p , ℓ q w , for sufficiently small sampling density δ > 0. Crucially, while the decomposition space D Q, L p , ℓ q w is defined using the bandlimited family Φ, the construction presented here usually allows for the prototype γ to be compactly supported in space. We emphasize that the theory presented here can cover the whole range p, q ∈ (0, ∞] and not only the case p, q ∈ [1, ∞] of Banach spaces.An important feature of our theory is that in many cases, the system Ψ δ will simultaneously form a Banach frame and an atomic decomposition for D Q, L p , ℓ q w . This implies that for frames of the form Ψ δ , analysis sparsity is equivalent to synthesis sparsity, i.e., the analysis coefficientsand only if f is an element of a certain decomposition space, if and only if f = i∈I,k∈Z d cThis is very convenient, since for many frame constructions-like shearlets-one only knows that the analysis coefficients for a class of "nice" signals are sparse. This, however, only entails synthesis sparsity with respect to the dual frame, about which often only limited knowledge is available. Using the theory presented here, one can derive synthesis sparsity with respect to the primal frame, for which one has an explicit formula and whose properties like smoothness and time-frequency localization are well understood.As a sample application, we show that the developed theory applies to α-modulation spaces and to (inhomogeneous) Besov spaces. In a companion paper, we also show that the theory applies to shearlet smoothness spaces.

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“…For example, by using the uncondition basis for α-modulation spaces constructed in [17], one can generate a compactly supported basis for the α-modulation spaces.…”

Section: Theorem 44mentioning

confidence: 99%

Compactly Supported Frames for Decomposition Spaces

Nielsen

Rasmussen

2011

J Fourier Anal Appl

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Abstract. In this article we study a construction of compactly supported frame expansions for decomposition spaces of Triebel-Lizorkin type and for the associated modulation spaces. This is done by showing that a finite linear combination of shifts and dilates of a single function with sufficient decay in both direct and frequency space can constitute a frame for Triebel-Lizorkin type spaces and the associated modulation spaces. First, we extend the machinery of almost diagonal matrices to Triebel-Lizorkin type spaces and the associated modulation spaces. Next, we prove that two function systems which are sufficiently close have an almost diagonal "change of frame coefficient" matrix. Finally, we approximate to an arbitrary degree an already known frame for Triebel-Lizorkin type spaces and the associated modulation spaces with a single function with sufficient decay in both direct and frequency space.

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Orthonormal bases for α-modulation spaces

Cited by 8 publications

References 19 publications

Structured, compactly supported Banach frame decompositions of decomposition spaces

Structured, compactly supported Banach frame decompositions of decomposition spaces

Structured, compactly supported Banach frame decompositions of decomposition spaces

Compactly Supported Frames for Decomposition Spaces

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