Tensor product Gauss-Lobatto points are Fekete points for the cube (Q2723527)

In this interesting paper, the authors show that Fekete points for the \(d\)-dimensional cube \([-1,1]^d\) are tensor products of the Fekete points in \([-1,1]\), which are known to be the Gauss-Lobatto quadrature points. The proof is not straightforward, so the result is of substantial intrinsic interest. The practical motivation for the problem comes from spectral element methods, where one searches for suitable collocation points. The result itself suggests that Fekete points may be useful for spectral element methods involving triangular elements. Recall that the Fekete points of order \(n\) for \([-1, 1]\) are points \(\{x_j\}^n_{j=0}\) in \([-1,1]\) such that the determinant \(\det (x_j^k)_{0\leq j,k\leq n}\) has maximal absolute value. It is an old result of Fejér that these Fekete points are the zeros of \((1-x^2) p'_n(x)\), where \(p_n\) is the \(n\)-th Legendre polynomial. To define a \(d\)-dimensional analogue, one must order the monomials in \(d\) variables. For each multi-index \({\mathbf i}= (i_1, i_2, \dots,i_d)\) of integers \(i_k\in [0,n]\), \(1\leq k\leq d\), and for \({\mathbf x}=(x_1,x_2, \dots,x_d)\), we define NEWLINE\[NEWLINEm_{\mathbf i}({\mathbf x})=x_1^{i_1} x_2^{i_2} \dots x_d^{i_d} \quad\text{and} \quad|{\mathbf i}|=\max_k i_k. NEWLINE\]NEWLINE There are \((n+1)^d\) such monomials. Given a set of \((n+1)^d\) points \(\{A_{ \mathbf j}\}\) contained in \([-1,1]^d\), we seek to maximize the absolute value of the determinant NEWLINE\[NEWLINE\det\bigl( m_{\mathbf i}({\mathbf A}_{\mathbf j})\bigr)_{|{\mathbf i}|, |{\mathbf j}|\leq n}.NEWLINE\]NEWLINE The main result is that the tensor product of the Gauss-Lobatto points (or Fekete) points for \([-1,1]\) maximizes this determinant. The paper is of interest to anyone studying or using multivariate approximation or spectral element methods.