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Sur l’approximation et la complétude des translatés dans les espaces de fonctions

Abstract : We are interested in the study of cyclicity and bicyclicity in weighted ℓ^p(Z) spaces and the study of cyclicity in Dirichlet spaces. While Wiener characterized the bicyclicity in ℓ¹(Z) and ℓ²(Z), thanks to the zero set of the Fourier transform, Lev and Olevski have shown that this set cannot characterize bicyclicity in ℓ^p(Z) when 1≺p≺2 for sequences ℓ¹(Z). Also Beurling, Salem and Newman were interested in the bicyclicity in ℓ^p(Z) when 1≺p≺2. In this work, we first extend the results of Beurling, Salem and Newman to the weighted ℓ^p(Z) spaces, by studying the Hausdorff dimension and the capacity of the zero set of the Fourier transform. Then we prove that the Lev-Olevskii result remains valid for cyclicity in ℓ^p(Z), 1≺p≺2. In addition, we give sufficient conditions for the cyclicity in the weighted ℓ^p(Z) spaces. Finally, we prove that, for a function f in the disk algebra and in a generalized Dirichlet space, if f is outer and the zero set of f is reduced to a point then f is cyclic. This generalizes the result of Hedenmalm and Shields who have treated the case of the classical Dirichlet space.
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Florian Le Manach. Sur l’approximation et la complétude des translatés dans les espaces de fonctions. Mathématiques générales [math.GM]. Université de Bordeaux, 2018. Français. ⟨NNT : 2018BORD0237⟩. ⟨tel-01962225⟩

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