New Classes of Radial Basis Functions for Quasi-Interpolation

Abstract

Radial basis functions (RBFs) are considered an important tool in numerical mathematics for approximating functions based on scattered data. They are often used as an interpolation method. In this process, a system of linear equations has to be solved. The size of the linear system is determined by the number of interpolation points. As the number of interpolation points increases, the solution tends to become numerically unstable, which results in the approximation of the function based on the scattered data becoming inaccurate. One possible approach to avoid this problem is provided by quasi-interpolation using RBFs. Quasi-interpolation is an efficient approximation method that does not require a system of linear equations to be solved. Just like interpolation with RBFs, quasi-interpolation is applied in arbitrary dimensions. Since many problems in industry and science are based on the approximation of scattered data, efficient approximation is regarded as a central task. With a rapidly increasing amount of available data, quasi-interpolation is gaining importance compared to classical interpolation. To compute the quasi-interpolation, a RBF has to be selected in advance. The approximation quality is affected by the selected RBF. The determination of an optimal RBF for a given approximation problem is considered an open research challenge. In this work, two new classes of RBFs are presented that are used to improve approximation by quasi-interpolation. The quality of an approximation is determined by the approximation order. A statement about the approximation order is provided by the Strang and Fix conditions. For their application, the asymptotic behavior of the distributional Fourier transform of the RBF is required. In addition to the two new classes of RBFs, a new class of non-RBFs is also introduced for use in quasi-interpolation. The analysis of approximation orders between RBFs and non-RBFs is fundamentally distinguished, and new approaches for further non-RBFs in quasi-interpolation are opened up. The results are shown to yield excellent convergence properties with low computational effort. The theoretical results presented are verified by numerical examples.