Asymptotics of the Christoffel functions on a simplex in \(\mathbb{R}^d\)
The author gives very advanced results on the Christoffel function \(\Lambda\) with respect to a weight \(W\), defined on a region \(\Omega\) or on the standard simplex \[ \Sigma^d=\{{\mathbf x}\in{\mathcal R}^d:x_1\geq 0,\ldots,x_d\geq 0,1-x_1-\cdots -x_d\geq 0\}. \] For a region \(\Omega\subset{\mathcal R}^d\), this function, given by \[ \Lambda_n(W;{\mathbf x})=\min_{P({\mathbf x})=1,P\in\Pi_n^d} \int_{\Omega} P^2({\mathbf y})W({\mathbf y}) d{\mathbf y}, \] plays an important role in the theory of orthogonal polynomials in several variables (it is the reciprocal of the reproducing kernel whose definition is independent of the choice of the orthonormal polynomial basis). First, in section 2, results are given for the Chebyshev weight \[ W_{\alpha}({\mathbf x})=w_{\alpha}x_1^{\alpha_1-1/2}\cdots x_d^{\alpha_d-1/2}(1-|{\mathbf x}|)^{\alpha_{d+1}-1/2},\;\alpha_i\geq 0, \] on the simplex \(\Sigma^d\) (the case \(\alpha_i=0\) only). In section 3 several very interesting results connecting various weights to \(W_0\) are derived. A nice paper.
orthogonal polynomials in several variables on a simplex
scientific article; zbMATH DE number 1333085