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 $ell^p(Z)$ spaces and the study of cyclicity in Dirichlet spaces. While Wiener characterized the bicyclicity in $ell^1(Z)$ and $ell^2(Z)$, thanks to the zero set of the Fourier transform, Lev and Olevski have shown that this set cannot characterize bicyclicity in $ell^p(Z)$ when $1 < p < 2$ for sequences in $ell^1(Z)$. Also Beurling, Salem and Newman were interested in the bicyclicity in $ell^p(Z)$ when $1 < p < 2$. In this work, we first extend the results of Beurling, Salem and Newman to the weighted $ell^p(Z)$ 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 $ell^p(Z)$, $1 < p < 2$. In addition, we give sufficient conditions for the cyclicity in the weighted $ell^p(Z)$ 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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