In this paper we generalize several results on uniform approximation orders with radial basis functions in (Buhmann, Dyn and Levin, 1993) and (Dyn and Ron, 1993) to L-approximation orders. These results apply, in particular, to approximants from spaces spanned by translates of radial basis functions by scattered centres. Examples to which our results apply include quasi-interpolation and least-...

We prove that the well known Lp-error estimates for radial basis function interpolation are optimal provided that the underlying function space is the native Hilbert space of the basis function. Furthermore we give upper bounds for the approximation orders in case of best L1-approximation using radial basis functions.

We give an explicit example for the selection of the shape parameter for a certain univariate radial basis function (RBF) interpolation problem.

where α = (α1, . . . , αn) is an n-tuple of nonnegative integers and |α| = ∑n j=1 |αj |. If every polynomial p ∈ πm−1(R) is determined by its values on X, then we will say that the data set X is unisolvent (for πm−1(R )). This condition can also be rephrased in terms of matrices. Order the monomials x in some convenient way. Form the matrix P for which the rows are an x evaluated at xj , j = 1,...

1 Radial Basis Functions 2 1.1 Multivariate Interpolation and Positive Definiteness . . . . . . 3 1.2 Stability and Scaling . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Solving Partial Differential Equations . . . . . . . . . . . . . . 7 1.4 Comparison of Strong and Weak Problems . . . . . . . . . . . 8 1.5 Collocation Techniques . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Method ...

A key problem in environmental monitoring is the spatial interpolation. The main current approach in spatial interpolation is geostatistical. Geostatistics is neither the only nor the best spatial interpolation method. Actually there is no “best” method, universally valid. Choosing a particular method implies to make assumptions. The understanding of initial assumption, of the methods used, and...

The local radial basis function (RBF) interpolation method enables very large-scale data sets to be handled efficiently, overcoming the drawbacks of global interpolation which produces highly ill-conditioned linear systems. Whereas there have been intensive studies on the accuracy of global RBF interpolation, the error analysis of local RBF interpolation is much less investigated. In this regar...

using the fixed point theorem in a‎ ‎cone‎, ‎the existence and multiplicity of radial convex solutions of‎ ‎singular system of monge-amp`{e}re equations are established‎.‎

using the fixed point theorem in a‎ ‎cone‎, ‎the existence and multiplicity of radial convex solutions of‎ ‎singular system of monge-amp`{e}re equations are established‎.‎