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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
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[1] Bloom, T, The Lebesgue constant for Lagrange interpolation in the simplex, J. approx. theory, 54, 338-353, (1988) · Zbl 0705.41002
[2] Bos, L.P, Bounding the Lebesgue function for Lagrange interpolation in a simplex, J. approx. theory, 38, 43-59, (1983) · Zbl 0546.41003
[3] Butzer, P.L; Hinsen, G, Two dimensional nonuniform sampling expansions—an iterative approch, I and II, Appl. anal., 32, 53-85, (1989) · Zbl 0646.94005
[4] Dyn, N, Interpolation and approximation by radial and related functions, (), 291 · Zbl 0705.41006
[5] \scH. G. Feichtinger and K. Grochenig, Reconstruction of band limited functions from irregular sampling values, preprint. · Zbl 0790.42021
[6] Gelfand, I.M; Vilenkin, N.Ya, ()
[7] Madych, W.R; Nelson, S.A, Multivariate interpolation and conditionally positive definite functions, Approx. theory appl., 4, No. 4, 77-89, (1988) · Zbl 0703.41008
[8] Madych, W.R; Nelson, S.A, Multivariate interpolation and conditionally positive definite function, II, Math. comp., 54, 211-230, (1990) · Zbl 0859.41004
[9] Morse, M; Cairns, S.S, Critical point theory in global analysis and differential topology, (1969), Academic Press New York · Zbl 0177.52102
[10] \scF. J. Narcowich and J. D. Ward, Norms of inverses and condition numbers for matrices associated with scattered data, preprint. · Zbl 0724.41004
[11] van der Waerden, B.L, ()
[12] Walker, R.J, Algebraic curves, (1962), Dover New York
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