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

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.

MSC:

41A05 Interpolation in approximation theory
41A63 Multidimensional problems
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