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Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. (English) Zbl 0764.41003
Let $$h$$ be a function on $$R^ n$$ which is conditionally positive definite of order $$m$$, let $$X=\{x_ j$$: $$1\leq j\leq N\}$$ be a set of points in $$R^ n$$, and let $$F=\{f_ j$$: $$1\leq j\leq N\}$$ be a set of real or complex numbers. The so-called $$h$$ spline interpolant for $$X$$ and $$F$$ is the function $s(x)=p(x)+\sum_{j=1}^ N c_ j h(x-x_ j),$ where $$p$$ is a polynomial of degree $$\leq m-1$$, and the $$c_ j's$$ are chosen so that both $$\sum_{j=1}^ N c_ j q(x_ j)=0$$ for all polynomials $$q$$ of degree $$\leq m-1$$, and $$s(x_ j)=f_ j$$, $$1\leq j\leq N$$. The main result can be described as follows. If $$f$$ is a continuous function and if $$X$$ and $$F$$ satisfy $$f(x_ j)=f_ j$$, and if certain natural restrictions are placed on $$f$$ and $$h$$, then, given a positive number $$b_ 0$$ there exist positive constants $$\delta_ 0$$ and $$\lambda$$, $$0<\lambda<1$$, such that if $$E$$ is a cube in $$R^ n$$ with side $$b\geq b_ 0$$, if $$0<\delta<\delta_ 0$$, and if each subcube of $$E$$ with side $$\delta$$ contains a point of $$X$$, then the spline interpolant for $$X$$ and $$F$$ satisfies $$| f(x)- s(x)|=O(\lambda^{1/\delta})$$ for all $$x\in E$$.
This improves an earlier result of the authors [Math. Comp. 54, 211-230 (1990)]. Other related results are also given. Using in the development of these results is a bound on the size of polynomial over a cube in $$R^ n$$ in terms of its values on a finite set of points which are spread out somewhat uniformly inside the cube.

##### MSC:
 41A05 Interpolation in approximation theory 41A10 Approximation by polynomials
##### Keywords:
$$h$$ spline interpolant
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##### References:
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