## 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

### Citations:

Zbl 0764.41003; Zbl 0004.05605
Full Text:

### References:

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