On best approximation of the monomials on the unit ball of \(\mathbb{R}^ r\)
The Kergin interpolant \({\mathcal K} (f)\) of a function \(f\in C^\mu (\mathbb{R}^r)\) with respect to \(\mu+1\) distinguished nodes is an unique polynomial in \(r\) variables of degree at most \(\mu\). [\textit{P. Kergin}, J. Approximation Theory 29, 278-293 (1980; Zbl 0492.41008)]. If \(f\) is a monomial \(x^m\), with \(|m|=\mu+1\), where \(m=(m_1, \dots, m_r) \in N^r\), then \({\mathcal K} (x^m)\) can be explicitly given, although for a general function \(f\in C^\mu (R^r)\) this fact is not possible. Bos and Liang have separately proved in the case \(r\geq 2\), that for a particular choice of the nodes, the Kergin interpolant \({\mathcal K} (x^n)\) is a best approximation of \(x^m\) in the uniform norm on the unit disk of \(\mathbb{R}^r\). In the present paper, sufficient conditions for \(\mu+1\) nodes are given in order that, in the case \(r=2\), the corresponding interpolant \({\mathcal K} (x^m)\) is a best approximation of \(x^m\) on the unit ball in the infinity norm and also, in the general case \(r\geq 3\), the corresponding interpolant \({\mathcal K} (x^m)\) is a best approximation of \(x^m\) on \(S^{r-1}\) with respect to the \(L_2\) norm. The given conditions make possible to get all the systems of nodes with the desired property.
scientific article; zbMATH DE number 1121221