Solvability of multivariate interpolation by radial or related functions (Q1801546)

Let \(X\) be a linear space and let \(x_ 1,\dots,x_ n\) be \(n\) points in \(X\). The paper deals with the question of interpolating given data at the nodes \(x_ i\), \(i=1,\dots,n\), by linear combinations of \(n\) functions on \(X\), of the form \(x\mapsto \sum^ m_{\nu=1} F_ \nu(\| T_ \nu(x- y_ k)\|^ 2)\), where the knots \(y_ k\in X\), \(F_ \nu\) are univalent real valued functions, and \(T_ \nu\) are linear mappings from \(X\) into a Hilbert space \(H\). Most of the paper is restricted to the case where \(x_ k= y_ k\), \(k=1,\dots,n\), i.e., the knots coincide with the nodes. In this case the interpolation problem is well-posed exactly when the interpolation matrix \(\Bigl\{\sum^ m_{\nu=1}F_ \nu(\| T_ \nu(x_ i- y_ k)\|^ 2)\Bigr\}^ n_{i,k=1}\), is nonsingular. The paper shows that for certain class of functions \(F_ \nu\) the nonsingularity of the matrix is related to the question whether the semi- norm \(| c|_{{\mathcal N},{\mathcal T}}:=\Bigl\{\sum^ m_{\nu=1} \sum_{y\in H} \Bigl(\sum_{\{i: T_ \nu x_ i=y\}} c_ i\Bigr)^ 2\Bigr\}^{1/2}\), where \(c:=(c_ 1,\dots,c_ n)\), is a norm on \(\mathbb{R}^ n\). Here \({\mathcal N}:=\{x_ 1,\dots,x_ n\}\) and \({\mathcal T}:=\{T_ \nu\}^ n_{\nu=1}\). Some sufficient conditions for the well-posedness are given also when the knots and nodes do not coincide.