For a xed conditionally positive deenite function we consider the linear space R containing all nite linear combinations of translates of augmented by a polynomial part. We equip R with a topology and provide characterizations for the largest space T among all linear spaces whose elements are continuous on R. Our approach serves to prove that all conditionally positive deenite functions are ava...

László T. Kóczy* **, Domonkos Tikk* ** and Tamás D. Gedeon** * Budapest University of Technology and Economics, Budapest, Hungary ** Murdoch University, Perth, W.A., Australia

In the context of radial basis function interpolation, the construction of native spaces and the techniques for proving error bounds deserve some further clari cation and improvement. This can be described by applying the general theory to the special case of cubic splines. It shows the prevailing gaps in the general theory and yields a simple approach to local error bounds for cubic spline int...

We present solutions for GPS orbit computation from broadcast and precise ephemerides using a group of artificial neural networks (ANNs), i.e. radial basis function networks (RBFNs). The problem of broadcast orbit correction, resulting from precise ephemerides, has already been solved using traditional polynomial and trigonometric interpolation. As an alternative approach RBFN broadcast orbit c...

In this paper, we propose a brief and general process to compute the eigenvalue of arbitrary waveguides using meshless method based on radial basis functions (MLM-RBF) interpolation. The main idea is that RBF basis functions are used in a point matching method to solve the Helmholtz equation only in Cartesian system. Two kinds of boundary conditions of waveguide problems are also analyzed. To v...

This invited contribution first reviews basic features of multivariate interpolation by radial basis functions, before selected of its advanced topics are addressed, including recent results concerning local polyharmonic spline interpolation. The latter is one main ingredient of a novel class of adaptive meshfree semi-Lagrangian methods for transport problems. The construction of these particle...

It is well-known that radial basis function interpolants suffer of bad conditioning if the basis of translates is used. In the recent work [12], the authors gave a quite general way to build stable and orthonormal bases for the native space NΦ(Ω) associated to a kernel Φ on a domain Ω ⊂ Rs. The method is simply based on the factorization of the corresponding kernel matrix. Starting from that se...

Interpolation by translates of a given radial basis function (RBF) has become a well-recognized means of tting functions sampled at scattered sites in R d. A major drawback of these methods is their inability to interpolate very large data sets in a numerically stable way while maintaining a good t. To circumvent this problem, a multilevel interpolation (ML) method for scattered data was presen...

There are multiple reasons why anisotropic basis functions may be needed or be more appropriate. The most obvious is that if the basis function is to be defined on Rn × T then there is no natural norm on this space that would reflect the unique properties of time. A second reason is that function being interpolated or approximated may incorporate a directional dependence. Thirdly, differentiabi...