Scattered cardinal Hermite interpolation by positive definite functions
Let \(E\subset C(R^d)\) be the subspace of all complex-valued continuous functions \(h\) in \(R^d\), such that for some \(\varepsilon>0\), \(\sup_{x\in R^d}|h(x)|(1+\|x\|_2)^{d+\varepsilon}<\infty\). Let \(\mathcal P:=\{p_1,\dots,p_r\}\) be a set of linearly independent homogeneous polynomials with complex coefficients and let \(h_1\dots,h_r\) be so that \(p_\mu h_\nu\in E\) for all \(1\leq\mu,\nu\leq r\). Finally, let \(J\subset R^d\)be countable. The paper discusses the Scattered Cardinal Hermite interpolation problem (SCHIP) with \(\mathcal P\) and \(J\), namely, given any \(r\) sequences \(d_k:=(d_{\mu,j})_{j\in J}\), \(\mu=1,\dots,r\), find a function of the form \(s(x)=\sum_{\nu=1}^r\sum_{j\in J}c_{\nu,j}h_\nu(x-j)\), so that \((p_\mu(D)s)(j)=d_{\mu,j}\), \(\mu=1,\dots,r\), \(j\in J\). The author calls the problem \(\mathcal L^p_r(J)\) solvable if there is a solution for any \(r\) sequences in \(l^p(J)\). He gives a characterization for a general \(1\leq p\leq\infty\), and in the case \(p=2\) describes it by means of the positive definiteness of certain matrices.
scientific article; zbMATH DE number 1471526