Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation.

*(English)*Zbl 0764.41003Let \(h\) 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.

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.

Reviewer: P.Lappan (East Lansing)

##### Keywords:

\(h\) spline interpolant
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\textit{W. R. Madych} and \textit{S. A. Nelson}, J. Approx. Theory 70, No. 1, 94--114 (1992; Zbl 0764.41003)

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