2009
DOI: 10.4064/sm192-1-6
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Molecules in coorbit spaces and boundedness of operators

Abstract: Abstract. We study the notion of molecules in coorbit spaces. The main result states that if an operator, originally defined on an appropriate space of test functions, maps atoms to molecules, then it can be extended to a bounded operator on coorbit spaces. For time-frequency molecules we recover some boundedness results on modulation spaces, for time-scale molecules we obtain the boundedness on homogeneous Besov spaces.1. Introduction. A remarkable principle of classical analysis states that an operator that … Show more

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Cited by 16 publications
(18 citation statements)
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“…The voice transform can be extended from the Hilbert space H to weighted Banach spaces of distributions using coorbit space theory. This theory was developed by Feichtinger and Gröchenig in a series of papers [14,15,16,18,20] and we collect the basic ideas in the following. Let now w be a real-valued, continuous and submultiplicative weight on G, i.e., w(gh) ≤ w(g)w(h) for all g, h ∈ G fulfilling in addition the conditions stated in [18,Section 2.2].…”
Section: Coorbit Spacesmentioning
confidence: 99%
“…The voice transform can be extended from the Hilbert space H to weighted Banach spaces of distributions using coorbit space theory. This theory was developed by Feichtinger and Gröchenig in a series of papers [14,15,16,18,20] and we collect the basic ideas in the following. Let now w be a real-valued, continuous and submultiplicative weight on G, i.e., w(gh) ≤ w(g)w(h) for all g, h ∈ G fulfilling in addition the conditions stated in [18,Section 2.2].…”
Section: Coorbit Spacesmentioning
confidence: 99%
“…As shown in [33,Section 4.2] the admissibility of the window h is implied by the classical "smooth molecule" conditions involving decay of derivatives and vanishing moments (see [23,24,25]). For example, any radial Schwartz function h with all moments vanishing is adequate.…”
Section: Convolution-dominated Operatorsmentioning
confidence: 99%
“…Precise decay conditions on Φ amount to diverse qualities of the set of molecules, such as number of derivatives, order of decay, and vanishing moments [51,54,61]. In light of (1.10), the molecule condition pertains to the locality of the sampling expansion (1.9), explaining its fundamental role in approximation theory: not only do the samples {W ψ f (λ) : λ ∈ Λ} characterize W ψ f , but the value W ψ f (b, a) can be approximated, up to a well-controlled error, by a finite subset of samples {W ψ f (λ) : λ −1 (b, a) ∈ U }, with U ⊂ G compact.…”
Section: Introductionmentioning
confidence: 99%
“…For example, when applied to Calderón's formula (1.1), coorbit theory does not produce an expansion (1.4) consisting of molecules (1.10), as, for example, [51,Theorem 1.5] does. The absence of an abstract notion of molecule was noted in [54], where coorbit molecules are formally introduced and their basic properties are studied. The question remained open, however, whether atomic decompositions consisting of molecules can be obtained in full generality.…”
Section: Introductionmentioning
confidence: 99%
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