## An estimate for multivariate interpolation.(English)Zbl 0558.41008

An estimate of the $$L^ p=L^ p(R^ n)$$ norm of a function is given in terms of its values on a (not necessarily regular) grid of points and the $$L^ p$$ norms of its k-th order derivatives, $$kp>n$$. The result is useful in obtaining estimates for multivariate interpolation schemes; this is illustrated by an example involving generalized splines.

### MSC:

 41A05 Interpolation in approximation theory 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 41A63 Multidimensional problems
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### References:

 [1] Adams, R.A, Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101 [2] Ahlberg, J.H; Nilson, E.N; Walsh, J.L, The theory of splines and their applications, (1967), Academic Press New York · Zbl 0158.15901 [3] Bramble, J.H; Hilbert, S.R, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. numer. anal., 7, 112-124, (1970) · Zbl 0201.07803 [4] Ciarlet, P.G; Raviart, P.A, General Lagrange and Hermite interpolation in Rn with applications to finite element methods, Arch. rational mech. anal., 46, 177-199, (1972) · Zbl 0243.41004 [5] Duchon, J, Splines minimizing rotation invariant semi-norms in Sobolev spaces, (), 85-100 [6] Fisher, S.D; Jerome, J.W, Elliptic variational problems in L2and L∞, Indiana univ. math. J., 23, 685-698, (1974) · Zbl 0271.41024 [7] Franke, R, Scattered data interpolation: tests of some methods, Math. comp., 38, No. 157, 181-200, (1982) · Zbl 0476.65005 [8] Meinguet, J, An intrinsic approach to multivariate spline interpolation at arbitrary points, (), 163-190 [9] Meinguet, J, A convolution approach to multivariate representation formulas, (), 198-210 [10] Meinguet, J, Sharp “a priori” error bounds for polynomial approximation in Sobolev spaces, (), 255-274 [11] Potter, E.H, Multivariate polyharmonic spline interpolation, () · Zbl 0558.41008 [12] Schumaker, L.L, Fitting surfaces to scattered data, (), 203-268 · Zbl 0343.41003
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