Generalised sk-spline interpolation on compact abelian groups
Let \(G\) be a compact abelian metric group, \(\widehat G\) the dual group of \(G\). Let be \(k\in C(G)\). Then an sk-spline is a funktion of the form \(c_0+\sum_{i=1}^{n}c_ik(\dot x_i^{-1})\), where \(n\in\mathbb N\), \(\sum_{i=1}^n c_i=0\), \(x_1,\dots,x_n\in G\). Density of the linear span of sk-splines in \(C(G)\) is proved. The authors also consider interpolation problems for these splines and the problem of the minimization of seminorm \(\int_G| (B^*)^{-1}g(x)| ^2d\mu(x)\), where \(B\) is a continuous linear operator from \((C(G))'\) to \(C(G)\). They also give an estimation of the approximation of convolution classes by sk-splines on \(\mathbb T^d\).
scientific article; zbMATH DE number 1278847
scientific article