Interpolation by periodic radial basis functions (Q1206832)

It is the aim of the paper to consider interpolation on a set of nodes \(y_ i\) on the unit circle \(S^ 1\) by radial basis functions of the form \(f(d(x,y_ j))\), where \(d\) denotes the geodesic distance on \(S^ 1\). For this purpose it is natural to investigate the behavior of the interpolation matrix \(A\) with entries \(A_{i,j}=f(d(y_ i,y_ j))\). The basic case, \(f(x)=x\), is discussed in more detail. Necessary and sufficient conditions on the nodes are obtained to ensure the invertibility of \(A\). For more general \(f\) the authors consider only the case of equidistant nodes \(y_ i\). Then the problem of invertibility of \(A\) is related to conditions on the Fourier coefficients of \(f\). This gives a nearly complete description of the existence of \(A^{-1}\) in this setting. Finally, a lot of interesting examples illustrate theses conditions.