2006
DOI: 10.1007/s00041-005-5041-6
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On Translation and Affine Systems Spanning L1(ℝ)

Abstract: We show that the discrete translation parameter sets ⊂ R for which some ϕ ∈ L 1 (R) exists such that the translates ϕ(x − λ), λ ∈ , span L 1 (R) are exactly the uniqueness sets for certain quasianalytic classes, and give explicit constructions of such generators ϕ. We also consider a similar situation for affine systems of the type ϕ(µx − λ), µ ∈ , λ ∈ .

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Cited by 9 publications
(14 citation statements)
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“…When p = 1 this atomic decomposition was found earlier by Bruna [5,Theorem 4]. Corollary 2 localizes the atomic decomposition to L p ( ), for domains ⊂ R d .…”
mentioning
confidence: 59%
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“…When p = 1 this atomic decomposition was found earlier by Bruna [5,Theorem 4]. Corollary 2 localizes the atomic decomposition to L p ( ), for domains ⊂ R d .…”
mentioning
confidence: 59%
“…For p = 1, the corollary was proved by Bruna [5,Theorem 4] for ψ ∈ L 1 with R d ψ dx = 0. His duality methods apply without our assumption that the translations lie in a lattice.…”
Section: Proposition 2 (Sufficient Conditions)mentioning
confidence: 87%
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“…Задача аффинного синтеза в пространстве L 1 (R d ) получила исчерпывающее решение в работах [2] и [3]. В [2] показано, что необходимым и достаточным условием положительного решения задачи аффинного синтеза при p = 1 явля-ется условие отличия от нуля интеграла от порождающей функции ψ:…”
Section: соответствующие теоремы представления устанавливаются на оснunclassified
“…See also [9] for a characterization of generating sets in L 1 (R) in terms of non quasianalytic classes of functions on R. The result of [10] was extended by Blank in [6] to the non quasianalytic Beurling algebras L 1 w (R) (see Sect. 4 for the definitions): We give a different and shorter proof of this result, using only the fact that L 1 w (R) is a regular Banach algebra and arguments similar to those employed in the proof of some of the results described in the previous section.…”
Section: Completeness Of Translates In Some Function Spaces On Rmentioning
confidence: 99%