Completeness of integer translates in function spaces on \(\mathbb{R}\)
The following problem of completeness is considered. Let \({\mathbf B}\) be a Banach space of functions (or distributions) on \(\mathbb{R}\) that is translation invariant, that is, \(T_sf\in{\mathbf B}\) when \(f\in{\mathbf B}\) and \(s\in\mathbb{R}\); \(T_sf(x)= f(x-s)\) stands for the translation operator. The problem is, does \({\mathbf B}\) contain a function \(f\in{\mathbf B}\) whose integer translates span the whole space, where span means the closed linear hull: \({\mathbf B}=\text{span} (T_nf:n\in\mathbb{Z})\)? Such functions are called generators or \(\mathbb{Z}\)-cyclic elements; functions \(f\) satisfying \({\mathbf B}=\text{span} (T_nf:n\geq 0)\) are called \(\mathbb{Z}_+\)-cyclic. The problem of completeness of all translates \(T_sf\), \(s\in\mathbb{R}\) is classical and well studied. For instance, the Wiener Tauberian theorem for the space \(L^1 (\mathbb{R})\), or generally, for the group algebra \(L^1(G)\) on a locally compact abelian group \(G\), describes all generators with respect to the group action in terms of their Fourier transforms. The existence in itself of such group generators is almost obvious for most, but not for all, translation invariant spaces. For integer translates, the situation is completely different, and, at first sight, the very fact of the existence of \(\mathbb{Z}\)-cyclic elements seems quite striking. For example, the basic spectral theory of unitary operators implies that there is no finite subset \(F\subset L^2(\mathbb{R})\) which would be \(\mathbb{Z}\)-cyclic in \(L^2(\mathbb{R})\). It is an easy exercise to see that the same is true for the space \(L^1(\mathbb{R})\). However, for many other natural function spaces, \(\mathbb{Z}\)-cyclic, and even \(\mathbb{Z}_+\)-cyclic, elements exist. For instance, the main result of the paper under review (theorem 2.1) says that each of the Banach spaces \(C_0^k(\mathbb{R})\), \(H^{p,k}(\mathbb{R})\), \(2<p<\infty\), \(k=0,1,\dots\) and \(L^2(\mathbb{R}, \rho(x)dx)\), where \(\rho\) is a tempered weight function in \(C_0(\mathbb{R})\), is singly generated. Here \(C_0^k(\mathbb{R})= \{f:f^{(j)}\in C_0(\mathbb{R})\), \(0\leq j\leq k\}\), \(H^{p,k}(\mathbb{R})= \{f:f^{(j)}\in L^p(\mathbb{R})\), \(0\leq j\leq k\}\). Moreover, each of these spaces has a generator \(\varphi\) verifying the following properties: (1) \(\varphi\in \bigcap_{k\geq 0}H^{2,k}(\mathbb{R})\), and \(\varphi\) is a restriction to \(\mathbb{R}\) of an entire function of a superexponential growth, and also satisfies one of the following conditions: (2) \(\varphi\) is already \(\mathbb{Z}_+\)-cyclic; (3) \(\{T_n\varphi:n\in \mathbb{Z}\}\) is an orthonormal system in \(L^2(\mathbb{R})\); (4) \(\{T_n\varphi:n\in \mathbb{Z}\}\) is a minimal system. It is proved that \(\varphi\) cannot be chosen to be the restriction to \(\mathbb{R}\) of an entire function of finite exponential type. The proof involves the use of uniqueness sets for the averaging (periodization) of \({\mathbf B}\) to the quotient group \(\mathbb{R}/\mathbb{Z}\) (theorem 2.3); for the case \({\mathbf B}=L^1(\mathbb{R})\) the mentioned periodization \(P\) is the classical one, \(P:L^1(\mathbb{R})\to L^1(\mathbb{R}/\mathbb{Z})\), \(Pf= \sum_{k\in\mathbb{Z}} T_{2\pi k}f\). The existence of the needed uniqueness sets is proved in section 3 (theorem 3.2 and proposition 3.3 for the spaces \(C_0^k(\mathbb{R})\), and theorems 3.4, 3.6, 3.8 and lemma 3.7 for the spaces \(H^{p,k}(\mathbb{R})\) and \(L^2(\mathbb{R}, \rho(x)dx))\). These results are illustrated by an example of a composition operator \(f(t)\mapsto f(t^2)\) on the spaces \(C_0[0,1]= \{f\in C[0,1]:f(0)= f(1)=0\}\) and \(L^p[0,1]\), which turns out to be cyclic (theorem 5.1). For results in the opposite direction, it is proved that the continuous embedding \({\mathcal F}{\mathbf B}\subset L_{\text{loc}}^1 (\mathbb{R})\) prevents from the existence of \(\mathbb{Z}\)-cyclic vectors, and even \(\mathbb{Z}\)-cyclic finite-dimensional subspaces (theorem 4.2). Reviewer's remarks. The cyclicity phenomenon for entire translations on function spaces was studied in \textit{N. Nikolski}'s papers [Sem. Math., V. A. Steklov Math. Inst., Leningrad 19(1970), 132-136 (1972); translation from Zap. Nauchn. Sem., Leningr. Otdel. Mat. Inst. Steklov 19, 227-236 (1970; Zbl 0258.47030) and Proc. Steklov Inst. Math. 120(1974), (1976); translation from Tr. Mat. Inst. Steklov 120 (1974; Zbl 0342.41027)]. The principal goal of these papers is to distinguish spaces having \(\mathbb{Z}\)- or \(\mathbb{Z}_+\)-cyclic vectors from spaces not possessing this property. All results of the paper under review, excepting particular properties (1)--(4) of cyclic vectors, are contained in the just cited papers. Moreover, a much more general setting is considered, namely the cyclic vectors problem for vector valued shift operators \(T_n:\ell(E)\to \ell(E)\), where \(\ell\) is an ideal sequence space (a lattice on \(\mathbb{Z}\)) and \(E\) is an arbitrary separable Banach space. The special cases considered in the paper under review correspond to the choice \(\ell=c_0(\mathbb{Z})\), \(E=C[0,1]\), or \(\ell=\ell^p(\mathbb{Z})\), \(E=W_p^k[0,1]\) (Sobolev spaces), or \(\ell=\ell^2(\mathbb{Z}, \rho(n))\), \(E=L^2[0,1]\), respectively. The techniques used for the proof are essentially the same. Besides these examples, many others are exhibited in the papers cited above, usually with necessary and sufficient conditions on the space parameters guaranteeing the existence of cyclic vectors. Namely, Orlicz, Lorentz, Marcinkiewicz, and general rearrangement invariant spaces are treated, as well as weighted versions of these spaces. Theorem 5.1 of the paper under review is contained in a result of \textit{A. K. Kitover}'s thesis [``Spectral analysis of isometric operators in spaces \(C\) and \(L^p\)'', Thesis, University of Leningrad, 1973]; namely, using the results of the papers cited above, the cyclicity is proved for composition operators \(C_\omega f=f\circ\omega\) and skew composition operators \(C_\omega f=M\cdot (f\circ\omega)\) for a general class of homeomorphisms \(\omega:K\to K\) of a metric compact \(K\) and a general class of functions \(M\in C(K)\).
scientific article; zbMATH DE number 980675