On approximation by a nonfundamental sequence of translates (Q1916835)

Let \(\{c_n\}\) be a sequence of distinct real numbers satisfying the separation condition \(||c_n|^p-|c_r|^p|\leq \rho|n-r|\), \(\rho>0\), for some integer \(p>0\). Let the functions \(f(t)\) and \(F(t)\) satisfy \(f(t)= O\{\exp(-\alpha t^2)\}\), \(F(t)= O\{\exp(- at^2)\}\) as \(|t|\to\infty\) and \(\exp(-bt^2)/F(t)\in L_2(\mathbb{R})\) for some positive numbers \(\alpha\), \(a\) and \(b\). The author proves the following. Theorem 1. Suppose \(S\) is the linear span of \(\{f(c_n- t)\}\) and \(g(t)\) is in the \(L_2(\mathbb{R})\) closure of \(S\). Then there exists a sequence \(\{b_n\}\) of real numbers such that \(g(t)=\sum b_nf(c_n- t)\) a.e. on \(\mathbb{R}\). Theorem 2. Let \(g(t)\) be a function in the \(L_2(\mathbb{R})\) closure of \(S\). Let \((A,B)\) be a bounded interval, \(g(t)\) be continuous on \((A,B)\), and \(d_n\) denote the uniform distance from \(g(t)\) to the span of \(\{f(c_r-t):r=0,1,\dots,n)\}\) in \((A,B)\). Then for any \(0<\delta<\alpha\), there is a positive number \(D\) (independent of \(n\) and \(g\)) such that \(d_n\leq D|g|_2\exp(-\delta\rho n)\).