Discrete translates in function spaces (Q722380)

This is a continuation of previous work by these authors \textit{A. Olevskii} and \textit{A. Ulanovskii}, [J. Fourier Anal. Appl. 10, No. 1, 93--104 (2004; Zbl 1071.42020)]. The main result of the paper states that if \(X\) is a Banach space over \(\mathbb{R}\) then there exists a Schwartz function \(\varphi\in S(\mathbb{R})\) such that \(\{\varphi(t-\lambda): \lambda\in\Lambda\}\) spans \(X\), where \(\Lambda\) is a uniformly discrete set, meaning that \(\Lambda=\{\lambda_n=n+a_n: 0<| a_n| <Cr^{| n|}, n\in\mathbb{Z}, 0<r<1, C>0\}\). Examples include all \(L^p\) spaces with \(p>1\) (but not \(p=1\)) and Sobolev spaces \(W_p^l\), \(p>1\), \(l\in\mathbb{N}\) on condition that \(S(\mathbb{R})\) is continuously and densely embedded and that no element of the dual space has a spectrum concentrated on \(\mathbb{Z}\). This is also used to prove under some technical conditions that a tempered distribution \(f=\varphi*S'(\mathbb{R})\) is zero if \(f|_\Lambda=0\). If one generator \(\varphi\) does not work, then \(X\) may be spanned by \(\Lambda\)-translates of two generators. For simplicity, the proofs are given for \(\mathbb{R}\), but they can be extended to \(\mathbb{R}^d\).