Some Erdős-Feldheim type theorems on mean convergence of Lagrange interpolation (Q1066023)

Given a weight function w(x) in \([-1,+1]\), let \(p_ n(w)\) denote the corresponding sequence of orthonormal polynomials. Let \(x_{kn}^{(w)}\) \((k=0,1,...,n+1)\) denote the set of nodes consisting of the zeros of \(p_ n(w)\) and the points \(-1,+1\). Let \(Q_{n+2}(f,w)\) denote the Lagrange interpolation polynomials associated with these nodes and a function f. The following result is proved: if \(w(x)=\sqrt{1-x^ 2}\) or \(w(x)=(1-x^ 2)^{3/2}\), then \[ \lim_{n\to \infty}\int^{+1}_{- 1}| Q_{n+2}(f,w,x)-f(x)|^ p\frac{dx}{\sqrt{1-x^ 2}}=0,\quad \forall p>0,\quad \forall f\in C[0,1]. \]