Remarks concerning completeness of translates in function spaces
In his previous papers [Sem. Math. V. A. Steklov, Math. Inst. Leningrad 19, 132-136 (1970; Zbl 0258.47030)] and [``Selected problems of weighted approximation and spectral analysis'', Trudy Math. Inst. Steklova 120 (1974) (in Russian), Proc. Steklov Math. Inst. 120 (1974; Zbl 0342.41028) (Engl. transl.)], the author considered certain translation invariant sequence spaces \(\ell(\mathbb{Z}, E)\) where \(E\) is an arbitrary separable Banach space. For example, \(\ell\) might denote \(\ell^p\) or \(c_0\). A vector \(x\in \ell(\mathbb{Z}, E)\) is called 1-cyclic (respectively 2-cyclic) if the closed linear span of its integer (respectively positive integer) translates is the whole space. In the second cited paper, he proved that \(\ell(\mathbb{Z},E)\) has a 1-cyclic (respectively 2-cyclic) vector if and only if the same is true of \(\ell(\mathbb{Z}, \mathbb{C}^2)\). Moreover, \(\ell(\mathbb{Z}, \mathbb{C}^2)\) has no 2-cyclic vector if its space of Fourier transforms is continuously embedded in the space of measurable functions on the circle. In the present paper, the author shows how these results translate readily into the language of completeness theorems for integer translates for functions on \(\mathbb{R}\). In particular, there exist functions \(f\) whose positive integer translates are complete in the spaces \(C^k_0(\mathbb{R})\), \(L^p(\mathbb{R})\), \(W^k_p(\mathbb{R})\) as well as in their weighted and vector-valued analogues.
space of measurable functions on the circle
scientific article; zbMATH DE number 1309603
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