On Grünwald interpolation (Q1297781)

Let \(-1<x_n<\dots<x_1<1\) be the zeros of the generalized Jacobi polynomials of degree \(n\) which are orthogonal with respect to the weight \(w(x):=g(x)(1-x)^\alpha(1+x)^\beta\) \((x\in(-1,1))\), where \(g>0\), \(g'\in \text{Lip} 1\) on \([-1,1]\) and \(\alpha,\beta\in(-1,0)\). Further let \[ G_n(f,x):=\sum^n_{k=1}f(x_k)\ell^2_k(x) \] be the Grünwald interpolation operator of \(f\) defined on \([-1,1]\). Here \(\ell_k\) denotes the Lagrange fundamental polynomials. The author proves that for \(f\in C[-1,1]\) \[ \lim_{n\to\infty} G_n(f,x)=f(x) \] uniformly on \([-1,1]\) if and only if \(f(-1)=f(1)=0\).