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 ∈ .
“…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.…”
“…Surjectivity of synthesis when ψ has nonzero integral was proved earlier in [5,Theorem 4] for p = 1, and even earlier in [28,Theorem 2], [29], for p ≥ 1. These works all proceed by studying the analysis operator (proving Tf ∞ ( p ) ≈ f p ) and then invoking duality; thus they provide no constructive method of synthesis like we provide in [10] and in this article.…”
Section: Remarks On L P For P ≥ 1 and On Hardy And Sobolev Spacesmentioning
“…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.…”
“…Surjectivity of synthesis when ψ has nonzero integral was proved earlier in [5,Theorem 4] for p = 1, and even earlier in [28,Theorem 2], [29], for p ≥ 1. These works all proceed by studying the analysis operator (proving Tf ∞ ( p ) ≈ f p ) and then invoking duality; thus they provide no constructive method of synthesis like we provide in [10] and in this article.…”
Section: Remarks On L P For P ≥ 1 and On Hardy And Sobolev Spacesmentioning
“…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
We study cyclicity of operators on a separable Banach space which admit a bicyclic vector such that the norms of its images under the iterates of the operator satisfy certain growth conditions. A simple consequence of our main result is that a bicyclic unitary operator on a Banach space with separable dual is cyclic. Our results also imply that if S : (a n ) n∈Z −→ (a n−1 ) n∈Z is the shift operator acting on the weighted space of sequences 2 ω (Z), if the weight ω satisfies some regularity conditions and ω(n) = 1 for nonnegative n, then S is cyclic if lim n→+∞ log ω(−n)/ √ n = 0. On the other hand one can see that S is not cyclic if the series n≥1 log ω(−n)/n 2 diverges. We show that the question of Herrero whether either S or S * is cyclic on 2 ω (Z) admits a positive answer when the series n∈Z log ||S n ||/(n 2 + 1) is convergent. We also prove completeness results for translates in certain Banach spaces of functions on R.123 294 E. Abakumov et al.
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