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Polynomial interpolation of minimal degree. (English) Zbl 0890.65008

Let \({\mathcal X}_N=\{x_1, \dots, x_N\}\) be a set of distinct points in \({\mathbb R}^d\). Denote by \(\pi^d\) the space of all polynomials in \(d\) variables, and by \(\pi_n^d \subset \pi^d\), the subspace of polynomials of total degree \(\leq n\). The Lagrange interpolation problem with respect to \({\mathcal X}_N\) is poised in a subspace \(\mathcal P \subset \pi^d\) if for any \(f: {\mathbb R}^d \to {\mathbb R}\) there exists a unique \( P \in {\mathcal P}\) such that \[ P(x_i)=f(x_i), \quad i=0,\dots,N. \tag{*} \] Generalizing the concept of least interpolation spaces introduced by C. de Boor and A. Ron [Constr. Approx. 6, No. 3, 287-302 (1990; Zbl 0719.41006)], the author studies the construction of minimal degree interpolation spaces \(\mathcal P\), i.e. satisfying the following conditions: 1) the Lagrange interpolation problem with respect to \({\mathcal X}_N\) is poised in \(\mathcal P\); 2) if \(\mathcal P \subset \pi_n^d\), then there is no subspace \({\mathcal P}' \subset \pi_{n-1}^d\) such that the Lagrange interpolation problem is poised in \({\mathcal P}'\); 3) if \(\mathcal P \subset \pi_n^d\) and \( f \in \pi_k^d\) with \(k \leq n\), then \(P\) satisfying (*) belongs to \( \pi_k^d\). For these spaces, a Newton-type interpolation method and a remainder formula are derived; several examples are discussed. A particular minimal degree interpolation space is studied which combines properties of least interpolation with the advantage of minimal memory consumption.

MSC:

65D05 Numerical interpolation
41A05 Interpolation in approximation theory
41A63 Multidimensional problems

Citations:

Zbl 0719.41006
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