Nonnegative trigonometric polynomials with fixed mean passing through given points (Q1277543)

Let \(\sigma_0>0\) and \(d=(d_1,\dots,d_n)\) with \(d_i\geq 0\), \(\psi_j= \exp(\varphi_j)\), \(\varphi_j \in R,\;X=Y^*Y\), when \(y_{kj}\) are coefficients of Lagrange interpolation polynomials with nodes \(\{\psi_j\}\). Then following assertions are equivalent: 1) there exists a trigonometric polynomial \(g\) of degree less than \(n\) such that \(g(\psi_j)=d_j\), \( j=1,\dots,n\), and with constant term \(g_0=\sigma_0;\) 2) if \(\lambda_1,\dots,\lambda_n \) are eigenvalues of \(X\) and \(P^{-1}XP\) is diagonal, then there exists \(\alpha\in C^n\), such that \(\sum^n_{j=1} \lambda_j\alpha_j\underline{\alpha_j}=\sigma_0\) and \(|\sum_k p_{kj}\alpha_k|^2=d_j\), \(j=1,\dots,n\).