In many areas of mathematics, science and engineering, from computer graphics to inverse methods to signal processing, it is necessary to estimate parameters, usually multidimensional, by approximation and interpolation. Radial basis functions are a powerful tool which work well in very general circumstances and so are becoming of widespread use as the limitations of other methods, such as least squares, polynomial interpolation or wavelet-based, become apparent. The author's aim is to give a thorough treatment from both the theoretical and practical implementation viewpoints. For example, he emphasises the many positive features of radial basis functions such as the unique solvability of the interpolation problem, the computation of interpolants, their smoothness and convergence and provides a careful classification of the radial basis functions into types that have different convergence. A comprehensive bibliography rounds off what will prove a very valuable work.
Radial basis function methods are modern ways to approximate multivariate
functions, especially in the absence of grid data. They have been known,
tested and analysed for several years now and many positive properties have
been identified. This paper gives a selective but up-to-date survey of several
recent developments that explains their usefulness from the theoretical point
of view and contributes useful new classes of radial basis function. We consider
particularly the new results on convergence rates of interpolation with radial
basis functions, as well as some of the various achievements on approximation
on spheres, and the efficient numerical computation of interpolants for very
large sets of data. Several examples of useful applications are stated at the
end of the paper.
For a radial.basis function ~p: ~ ~ ~ we consider interpolation on an infinite regular lattice Ihf(x)= ~r,,f(kh)x(x/h-k), xe~", to f: ~'--, ~, where h is the spacing between lattice points and the cardinal function x(x)-~ kE ~" cA~(llx-ktl), x~", satisfies X(J)= 8oj for all j ~ ~". We prove existence and uniqueness of such cardinal functions X, and we establish polynomial precision properties of Ih for a class of radial-basis functions which includes ~(r)=r 2q+t, ~(r)---r 2q log r, ~(r)= r2,7/~c 2, and ~p(r)= 1/r2~c 2 where q E ~+. We also deduce convergence orders of lhf to sufficiently differentiable functions f when h --~ 0.
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