On certain configurations of points in \(\mathbb{R}{}^ n\) which are unisolvent for polynomial interpolation (Q1174324)

Let \(\mathcal P_ d(E)\) be the vector space of polynomials of degree \(d\) on a set \(E\subset R^ n\). Denote the dimension of \(\mathcal P_ d(E)\) by \(N_ d(E)\). The following polynomial interpolation problem is considered: given \(N_ d(E)\) points \(x_{\alpha}\in E\) and \(N_ d(E)\) function values \(f_{\alpha}\), to find a polynomial \(P\in\mathcal P_ d(E)\) such that \(P(x_{\alpha})=f_{\alpha}\), for all \(\alpha\). It is said that \(X:=\{x_ 1,\dots,x_{N_ d(E)}\}\) is unisolvent if the interpolation problem has a unique solution for all given sets of function values. Let \(\{q_ 1,\dots,q_{N_ d(E)}\}\) be a basis for \(\mathcal P_ d(E)\). Then for \(X\) the generalised Vandermonde determinant \[ VDM^ d_ E(X):=\det[q_{\alpha}(x_{\beta})]_{1\leq\alpha,\beta\leq N_ d(E)} \] is introduced. Some configurations of points on algebraic sets are described for which the polynomial interpolation problem has the property of unisolvency. Formulae for corresponding Vandermonde determinants and an illustrative example in \(R^ 2\) are presented. Presented results generalise some of the results of Chung and Yao and Gasca and Maeztu about Lagrange interpolation at points on hyperplanes and the formulae given by Chui and Lai for Vandermonde determinant in this special case.