Mean convergence of Grünwald interpolation operators (Q1811970)

Let \(w\) be a weight function on \([-1,1]\), and let \(X\) be the array of roots of the orthogonal polynomials corresponding to it. Let \[ l_{kn}(x):=\frac{w_n(x)}{(x-x_{kn})w'_n(x_{kn})},\quad 1\leq k\leq n=1,2,\dots, \] where \(w_n(x):=\prod_{k=1}^n(x-x_{kn})\). Define the Grünwald interpolation polynomials of \(f\in C[-1,1]\), by \[ G_n(f,x):=\sum_{k=1}^nf(x_{kn})l^2_{kn}(x). \] Among the results the paper proves that \[ \lim_{n\to\infty}\|G_n(f,x)-f\|_{w,1}, \] if and only if \[ \lim_{n\to\infty}\|\sum_{k=1}^nl^2_{kn}(x)-1\|_{w,1}=0, \] where the notation \(\|\cdot\|_{w,1}\) means the weighted \(L_1\) norm.