A Remark on The Geometry of Spaces of Functions with Prime Frequencies |
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Authors: | P Lefèvre E Matheron O Ramaré |
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Institution: | 1. Univ Lille-Nord-de-France UArtois, Laboratoire de Mathématiques de Lens EA 2462, Fédération CNRS Nord-Pas-de-Calais FR 2956, F-62 300, Lens, France 2. Univ Lille-Nord-de-France USTL, Laboratoire Paul Painlevé U.M.R. CNRS 8524, F-59 655, Villeneuve d’Ascq Cedex, France
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Abstract: | For any positive integer r, denote by \({\mathcal{P}_{r}}\) the set of all integers \({\gamma \in \mathbb{Z}}\) having at most r prime divisors. We show that \({C_{\mathcal{P}_{r}}(\mathbb{T})}\) , the space of all continuous functions on the circle \({\mathbb{T}}\) whose Fourier spectrum lies in \({\mathcal{P}_{r}}\) , contains a complemented copy of \({\ell^{1}}\) . In particular, \({C_{\mathcal{P}_{r}}(\mathbb{T})}\) is not isomorphic to \({C(\mathbb{T})}\) , nor to the disc algebra \({A(\mathbb{D})}\) . A similar result holds in the L 1 setting. |
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