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We generate various new radial kernels by taking derivatives of known kernels with respect to scale. This is different from the well-known scale mixtures used before. In addition, we provide a simple recipe that explicitly constructs new kernels from the negative Laplacian of known kernels. Surprisingly, these two methods for generating new kernels can be proven to coincide for certain standard classes of radial kernels. The resulting radial kernels are positive definite, and a few illustrations are provided.

In this paper, we deal with two different problems. First, we provide the convergence rates of multiresolution approximations, with respect to the supremum norm, for the class of elliptic splines defined in Ref. 10, and in particular for polyharmonic splines. Secondly, we consider the problem of recovering a function from a sample of noisy data. To this end, we define a linear and smooth estimator obtained from a multiresolution process based on polyharmonic splines. We discuss its asymptotic properties and we prove that it converges to the unknown function almost surely.

In this work, we study a method to recover surfaces with some discontinuity curves from a set of noisy data. By function with singularities, we mean that the function or its partial derivatives are discontinuous across planar curves. The method we present is well suited for correlated data on trajectories (i.e., the data are collected on given tracks as in the case for measurements gathered from ships or aircraft). We propose a method based on wavelet decomposition and splines as well. (C) 2000 Elsevier Science Ltd. All rights reserved

In this paper, we propose a method belonging to the class of special meshless kernel techniques for solving Poisson problems. The method is based on a simple new construction of an approximate particular solution of the nonhomogeneous equation in the space of polyharmonic splines. High order Lagrangian polyharmonic splines are used as a basis to approximate the nonhomogeneous term and a closed-form particular solution is given. The coefficients can be computed by convolution products of known vectors. This can be done in all dimensions, without numerical integration nor solution of systems. Numerical experiments are presented in two dimensions and show the quality of the approximations for different test examples

This paper applies difference operators to conditionally positive definite kernels in order to generate kernel $$B$$ -splines that have fast decay towards infinity. Interpolation by these new kernels provides better condition of the linear system, while the kernel $$B$$ -spline inherits the approximation orders from its native kernel. We proceed in two different ways: either the kernel $$B$$ -spline is constructed adaptively on the data knot set $$X$$ , or we use a fixed difference scheme and shift its associated kernel $$B$$ -spline around. In the latter case, the kernel $$B$$ -spline so obtained is strictly positive in general. Furthermore, special kernel $$B$$ -splines obtained by hexagonal second finite differences of multiquadrics are studied in more detail. We give suggestions in order to get a consistent improvement of the condition of the interpolation matrix in applications.

Under very mild additional assumptions, translates of conditionally positive definite radial basis functions allow unique interpolation to scattered multivariate data, because the interpolation matrices have a symmetric and positive definite dominant part. In many applications, the data density varies locally according to the signal behaviour, and then the translates should get different scalings that match the local data density. Furthermore, if there is a local anisotropy in the data, the radial basis functions should possibly be distorted into functions with ellipsoids as level sets. In such cases, the symmetry and the definiteness of the matrices are no longer guaranteed. However, this brief note is the first paper to provide sufficient conditions for the unique solvability of such interpolation processes. The basic technique is a simple matrix perturbation argument combined with the Ball-Narcowich-Ward stability results. © Springer-Verlag 2004.