Approximation by translates of a positive definite function (Q1922928)

The purpose of the paper is to investigate density properties of translates of a single function. Denote by \(\mathcal P\) the set of all positive definite functions on \(\mathbb{R}^n\) such that \(\sum^m_{i,j=1} c_ic_j f(x_i-x_j)\geq 0\) whenever \(x_1,\dots,x_m\in\mathbb{R}^n\) and \(c_1,\dots,c_m\in\mathbb{R}\). Let \(\Omega\) be a compact subset of \(\mathbb{R}^n\). The following statements are proved. Proposition 1. Let \(h\in{\mathcal P}\cap C^m(\mathbb{R}^n)\). Assume that the Fourier transform \(\widehat h\) is a locally integrable function, and that there is a set \(E\) of positive Lebesgue measure such that \(\widehat h(x)\neq 0\) for all \(x\in E\). Then the set \(\text{span}\{h(\cdot-y):y\in \Omega\}\) is dense in \(C^r(\Omega)\), where \(2r\leq m\). Proposition 2. Let \(h\in {\mathcal P}\cap C^m(\mathbb{R}^n)\). Assume that \(h\in L^p(\mathbb{R}^n)\) for some \(p\in[1,2]\) and \(|h|_p>0\). Then the set \(\text{span}\{h(\cdot-y):y\in \Omega\}\) is dense in \(C^r(\Omega)\), for \(2r\leq m\).