Chebyshev's maximum principle in several variables. (Q1404938)

Let \(\Pi_n^d\) be the space of polynomials of total degree not larger than \(n\) in \(d\) variables. Let \(L\) be a positive definite linear functional, that is \(L(p^2)>0,\) for all \(p\in\Pi_n^d\), \(n\in\mathbb N,p\neq 0.\) Furthermore, let \(\{P_{\alpha}^n\}_{| \alpha| =n}\) be a sequence of orthonormal polynomials with respect to \(L.\) The functional \(L\) is called Gaussian if \(\{P_{\alpha}^n\}\) has \(N=\dim\Pi_{n-1}^d\) common zeros \(X_1^n,\ldots,X_N^n.\) The authors prove the following generalization of a result of Chebyshev: if \(L\) is Gaussian and \(l\) is an arbitrary linear function on \(\mathbb R^d,\) then \[ \max_{1\leq i\leq N}l(X_i^n)=\max_{\pi_{n-1}\in\Pi_{n-1}^d,\pi_{n-1}\neq 0}\frac{L(l(x)\pi_{n-1}^2(x))}{L(\pi_{n-1}^2(x))}. \]