Necessary conditions for \(L_ p\) convergence of Lagrange interpolation on an arbitrary system of nodes (Q1914793)

Let \(d\alpha\in S\) (the Szegö's class) be a measure supported in \([-1;1]\). A necessary condition on a triangular matrix of nodes \(1\geq x_{1n}> \dots x_{nn} \geq -1\) for mean convergence in \(L_p (d\alpha)\), \(0< p< \infty\), of Lagrange interpolation for every \(f\in C [-1; 1]\) are given in terms of \(\omega_n (x)= (x- x_{1n}) \dots (x- x_{nn})\). Polynomials \(\omega_n (x)= P_n (d\alpha, x)/ \gamma_n\) with \(P_n\) being the orthonormal polynomial with respect to \(d\alpha\) satisfy those conditions.