In this study, we are mainly interested in error estimates of interpolation, using smooth radial basis functions such as multiquadrics. The current theories of radial basis function interpolation provide optimal error bounds when the basis function φ is smooth and the approximand f is in a certain reproducing kernel Hilbert space Fφ. However, since the space Fφ is very small when the function φ...

We study p-sets and (i)-p-sets for a real function space A and compare them with p-sets for its complexification. We also discuss the relation of p-sets and (i)-p-sets with weak peak sets and weak (i)-peak sets which generalize our result of [7]. Using some of the results, we generalize a result of Izzo [2] (proved for a complex function algebra) for a real function space A. Mathematics Subject...

Zernike polynomials provide a well known, orthogonal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. A related set of orthogonal functions is given here which represent vector quantities, such as mapping distortion or wavefront gradient. Previously, we have developed a basis of functions generated from gradients of Zern...

Small peptides are model molecules for the amino acid residues that are the constituents of proteins. In any bottom-up approach to understand the properties of these macromolecules essential in the functioning of every living being, to correctly describe the conformational behaviour of small peptides constitutes an unavoidable first step. In this work, we present an study of several potential e...

In part I we discussed the solution of corneal curvature using a 2D meshless method based on radial basis functions (RBFs). In Part II we use these methods to fit a full nonlinear thin membrane model to a measured data-set in order to generate a topological mathematical description of the cornea. In addition, we show how these results can lead to estimations for corneal radius of curvature and ...

Radial basis function networks (RBF) are efficient general function approximators. They show good generalization performance and they are easy to train. Due to theoretical considerations RBFs commonly use Gaussian activation functions. It has been shown that these tight restrictions on the choice of possible activation functions can be relaxed in practical applications. As an alternative differ...

The basis set superposition error-free second-order Møller-Plesset perturbation theory of intermolecular interactions, based on the "chemical Hamiltonian approach," which has been introduced in Part I, is applied here to open-shell systems by using a new, effective computer realization. The results of the numerical examples considered (CH(4) em leader HO, NO em leader HF) showed again the perfe...

The behavior of Radial Basis Function (RBF) Networks greatly depends on how the center points of the basis functions are selected. In this work we investigate the use of instance reduction techniques, originally developed to reduce the storage requirements of instance based learners, for this purpose. Five Instance-Based Reduction Techniques were used to determine the set of center points, and ...