An estimate for multivariate interpolation. II. (English) Zbl 1119.41003

Madych, W. R.

J. Approximation Theory 142, No. 2, 116-128 (2006).

The author is concerned with the estimation of the \(L^p\) norm of a function \(u\) (with enough regularity) on a domain of \(\mathbb{R}^n\) from the values of \(u\) on a certain discrete subset of the domain and the \(L^p\) norms of the distributional derivatives of \(u\) of a given order \(m\). Let \(u\) be an \(m\) times continuously differentiable function on a cube \(I\subset\mathbb{R}^n\). Partition \(I\) into a high enough number of equal subcubes and select a point from the interior of each subcube, thus obtaining a (possibly irregular) grid \(\mathcal{X}_0\). A key step in this paper is to estimate \(\| u\| _p\) in terms of the maximum of \(| u| \) on \(\mathcal{X}_0\) and the \(L^p\) norms of the derivatives of \(u\) of order \(m\). The proof relies on the corresponding result for polynomials of degree less than \(m\) (i.e., the \(m\)-derivatives vanish) from W. R. Madych and S. A. Nelson [J. Approximation Theory 70, No. 1, 94–114 (1992; Zbl 0764.41003)] and a multivariate version of Kowalewski’s remainder formula for Lagrange interpolation [G. Kowalewski, Interpolation und genäherte Quadratur, Leipzig, Berlin: B. G. Teubner (1932; Zbl 0004.05605)] established in the previous section. If \(m\), \(n\), and \(p\) satisfy the Sobolev Theorem conditions for imbedding in \(C^0\), a mollifier argument extends the estimate to Sobolev functions. The author focus then our attention on domains of \(\mathbb{R}^n\) which allow a covering by cubes of a certain size with controlled overlapping. The main result, (the) Theorem (in subsection) 3.5, estimates \(\| u\| _p\) on such domains in terms of the \(\ell^p\) norm of the values of \(u\) on an adequate discrete subset and the \(L^p\) norms of the derivatives of \(u\) of order \(m\). In the last section, Theorem 3.5 is applied to the estimation of \(\| u\| _q\) when \(q\neq p\). Some remarks are made about the applicability of these methods to more general domains.

Reviewer: Antonio Serra (Lisboa)

MSC:

41A05	Interpolation in approximation theory
41A63	Multidimensional problems

Keywords:

multivariate interpolation; Sobolev; Kowalewski’s remainder formula

Citations:

Zbl 0764.41003; Zbl 0004.05605

References:

[1]	Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[2]	Davis, P.J., Interpolation and approximation, (1975), Dover New York · Zbl 0111.06003
[3]	Madych, W.R.; Nelson, S.A., Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation, J. approx. theory, 70, 1, 94-114, (1992) · Zbl 0764.41003
[4]	Madych, W.R.; Potter, E.H., An estimate for multivariate interpolation, J. approx. theory, 43, 2, 132-139, (1985) · Zbl 0558.41008
[5]	Narcowich, F.J.; Ward, J.D.; Wendland, H., Sobolev bounds on functions with scattered zeros with applications to radial basis function surface Fitting, Math. comp., 74, 250, 743-763, (2005) · Zbl 1063.41013
[6]	Pesenson, I., A reconstruction formula for band limited functions in \(L^2(\mathbb{R}^d)\), Proc. amer. math. soc., 127, 12, 3593-3600, (1999) · Zbl 0998.42024
[7]	Wendland, H., Scattered data approximation, (2005), Cambridge University Press Cambridge · Zbl 1075.65021
[8]	Wendland, H.; Rieger, C., Approximate interpolation with applications to selecting smoothing parameters, Numer. math., 101, 643-662, (2005)