On the convergence of cardinal interpolation by parameterized radial basis functions
From the authors' abstract: ``In this article, we extend the results by relaxing the requirements on the approximand functions from several points of view. The radial basis functions that we are concerned with and which are of special interest contain the celebrated multiquadrics, inverse multiquadrics and shifted thin-plate spline radial basis functions for instance. We also generalize the classes of admitted approximands as well as the radial basis functions to generalized multiquadrics in place of the well-known ordinary or for example inverse multiquadrics. An interesting analytical aspect of this work is that -- unlike the classical Whittaker-Shannon theorem -- functions (approximands) may be reproduced for the parameter \(c\rightarrow\infty\) in the generalized multiquadrics cardinal approximands, where the usual Shannon series does not converge with these approximands due to the slow decay of the sinc-function which does not allow e.g. polynomials as approximands. In contrast to the latter, the generalised multiquadrics cardinal functions employed here decay sufficiently fast for each fixed parameter \(c\) that even polynomials may be admitted as approximands and are reproduced when then the parameter tends to infinity.''
scientific article; zbMATH DE number 6675615
scientific article