Will Light

Some Aspects of Radial Basis Function Approximation

This paper deals with three basic aspects of radial basis approximation. A typical example of such an approximation is the following. A function f in C (ℝn) is to be approximated by a linear combination of ‘easily computable’ functions g 1,…, g m For these functions the simplest choice in the radial basis context is to define g i by x ↦ ∥x − x i∥2 for x ∈ ℝn and i = l,2,…,m. Here ∥ · ∥2 is the usual Euclidean norm on ℝn. These functions are certainly easily computable, but do they form a flexible approximating set? There are various ways of posing the question of flexibility, and we consider here three possible criteria by which the effectiveness of such approximations may be judged. These criteria are labelled density, interpolation and order of convergence in the exposition.

Interpolation by piecewise-linear radial basis functions

Abstract In the two-dimensional plane, a set of nodes x 1 , x 2 , …, x n is given. It is desired to interpolate arbitrary data given at the nodes by a linear combination of the functions h i ( x ) = ∥ x − x i ∥. Here the norm is the l 1 -norm. For this purpose, one can employ the space PL of all continuous piecewise-linear functions on the rectangular grid generated by the nodes. Interpolation at the nodes by this larger space is quite easy. By adding an appropriate PL -function that vanishes on the nodes, we can obtain the linear combination of h 1 , h 2 , …, h n that interpolates the data. This algorithm is much more efficient than the straightforward method of simply solving the linear system of equations ∑ c j h j ( x i ) = d i .

Interpolation by periodic radial basis functions

Abstract On the unit circle S1, let d be the natural (geodesic) metric. We investigate the possibility of interpolating arbitrary data on a set of nodes yi ϵ S1 by means of a function of the form x ↦ ∑i = 1n cif(d(x, yi)). Here f is a function from [0, π] to R , and is subject to our choice. The interpolation matrix A having elements Aij = f(d(yj, yi)) is crucial to this problem. In the basic case, f(x) = x, we give necessary and sufficient conditions on the nodes for the invertibility of A. For equally-spaced nodes, we give nearly complete conditions on f for the invertibility of A.

Spaces of distributions, interpolation by translates of a basis function and error estimates

Summary. Interpolation with translates of a basis function is a common process in approximation theory. The most elementary form of the interpolant consists of a linear combination of all translates by interpolation points of a single basis function. Frequently, low degree polynomials are added to the interpolant. One of the significant features of this type of interpolant is that it is often the solution of a variational problem. In this paper we concentrate on developing a wide variety of spaces for which a variational theory is available. For each of these spaces, we show that there is a natural choice of basis function. We also show how the theory leads to efficient ways of calculating the interpolant and to new error estimates.

Constructive methods of approximation by ridge functions and radial functions

Let ϕ be a univariant function, and letg(x) be the average of ϕ(〈x,u〉) asu runs over the unit sphere in ℝn. We give a necessary and sufficient condition forg to be a kernel function, i.e., thatg be inL1 (ℝn) and have integral 1. The result is used to give a constructive proof of the density of the ridge functions based upon the function ϕ.

Error Estimates for Approximation by Radial Basis Functions

This paper considers the problem of error estimates for interpolation by radial basis functions. To this end, a recap of the theory of bounding linear functionals in Hilbert spaces is presented. We begin with a normed linear space X. Let γ 1,…, γ m be linear ‘information functionals’ on X such that, for a given f ∈ X, the ‘information’ γ i (f) = α i , i = 1,…, m is known. With this data, we wish to compute γ(f) where γ is another linear functional on X. By taking a set of distinct points A = {a 1,…, a m } and choosing the information functionals to be point evaluations, that is, γ i (f) = f(a i ), i = 1,…, m with γ(f) = f(x) for some fixed point x, we obtain a general interpolation problem. We will, however, concentrate on interpolation by radial basis functions and it will become clear that the analysis which leads to the error estimates can subsequently be used to characterise the interpolant itself. Thus, the theory presented here is a very powerful technique in interpolation theory.

Recent developments in the strang-fix theory for approximation orders

Abstract This paper deals with the approximation order that can be achieved by using dilates of integer translates of a finite number of compactly supported or rapidly decreasing functions. The function being approximated is assumed to be in some appropriate Sobolev space, and approximation is carried out by dilations of the integer translates. If h is the dilation parameter, then the object is to determine when the approximation scheme gives O(hk) rate of convergence.

A Simple Approach to the Variational Theory for Interpolation on Spheres

In this paper we consider the problem of developing a variational theory for interpolation by radial basis functions on spheres. The interpolants have the property that they minimise the value of a certain semi-norm, which we construct explicitly. We then go on to investigate forms of the interpolant which are suitable for computation. Our main aim is to derive error bounds for interpolation from scattered data sets, which we do in the final section of the paper.

Convolution operators for radial basis approximation

We construct a large class of continuous integrable functions on

Techniques for generating approximations via convolution kernels

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