On Wu and Schaback’s Error Bound

Radial basis functions are a very powerful tool in multivariate approximation since they are mesh free. In the theory of radial basis functions, the most frequently used error bounds are the one raised by Madych and Nelson in [5] and the one raised by Wu and Schaback in [6], especially the latter. It seems that Wu and Schaback’s error bound is five times more frequently used than that of Madych and Nelson’s. The reason is that Wu and Schaback’s space of approximated functions is much easier to understand, and their error bound is much easier to use. Unfortunately, this error bound contains crucial mistakes which will be pointed out in this paper. Moreover, the r.b.f. people have misunderstood its space of approximands for long. This will also be discussed in depth. These, to some extent, may be a blow to r.b.f. people. However, Madych and Nelson’s error bound is still very powerful because it’s highly related to Sobolev spaces which contain solutions of many differential equations. All these discussions are based on the central idea of this paper,i.e. unsmooth functions should be approximated by unsmooth functions.