Resumen de Interpolation and approximation by delsarte translates of radial basis functions
Cristian Arteaga Clemente
In approximation theory, radially symmetric, (conditionally) positive definite functions, or radial basis functions (RBFs), are used to solve scattered data interpolation problems in euclidean space. The setting for a variational approach to such interpolation problems, the so-called native spaces, was constructed by several authors upon seminal work of Micchelli and of Madych and Nelson in the late 1980's. A decade later, building upon previous work by Duchon, Light and Wayne ideated an alternative approach in which the distributional theories of the Fourier transformation and the Fourier convolution play a prominent role.
However, a multivariate radial function can be identified with a function on the nonnegative real axis, and its Fourier transform (which is, in turn, radial) coincides with a Hankel transform. Moreover, the natural convolution structure in the nonnegative real axis is given by (a suitable variant of) the convolution associated to the Hankel transformation, for which a distributional theory is available as well. This convolution is constructed upon a generalized translation operator, which (under minor variants) is known as Bessel, Delsarte or Hankel translation.
The present thesis exploits these ideas to establish the validity of new RBF interpolation and approximation schemes where the Hankel/Delsarte translation replaces the usual one. It consists of three main chapters.
In Chapter 1 we develop Light and Wayne approach to RBF interpolation, with the Hankel transformation and the Hankel convolution successfully replacing the Fourier ones. In this context, an estimate for the pointwise error of the interpolants is obtained. The study of Hankel conditionally positive definiteness (connecting our interpolation spaces with the standard native spaces) is initiated, and some examples of basis functions are given. Numerical experiments are appended.
The new interpolation spaces are endowed with a seminorm which involves a weight and is expressed in terms of the Hankel transform of each function (indirect seminorm). In Chapter 2 we discuss essentially two special classes of weights (so-called of type I and type II) that allow to write these indirect seminorms in direct form, that is, in terms of the function itself rather than its Hankel transform, thus leading to more convenient error estimates. Fairly general conditions, which are shown to be satisfied by type II weights, are formulated in order to guarantee the density in the interpolation spaces of some other spaces arising naturally in the theory. Finally, we obtain explicit representations of the basis functions, the members of the interpolation spaces and their higher-order Bessel derivatives, along with some polynomial bounds for them.
A particularly interesting outcome of standard RBF interpolation concerns approximation by RBF neural networks (RBFNNs), which constitute a central theme in neurocomputing as well as in many other areas as diverse as finance, medicine, biology, geology, engineering and physics. In Chapter 3 we prove that the family of RBFNNs which results from replacing the usual translation with the Delsarte one, and taking the same smoothing factor in all kernel nodes, is dense (in neurocomputing terminology: has the universal approximation property) in spaces of weighted p-integrable functions and in spaces of continuous functions, endowed with certain natural topologies. Furthermore, a characterization of those kernels yielding universal approximation is obtained for p=1 and p=2, provided that the smoothing factor is allowed to vary across the kernel nodes. We end by solving the closely related question of obtaining analogues of the celebrated Wiener and Wiener-Pitt tauberian theorems for the Delsarte translation and the Fourier-Bessel transformation.
