A Hilbert transform representation of the error in Lagrange interpolation (Q1876666)

Let \(P_n\) be a polynomial of degree \(n\) with distinct real zeros, and given a function \(f\), let \(L_n[f]\) denote the Lagrange interpolation polynomial to \(f\) at the zeros of \(P_n\). In this note, the authors give a representation of the error \(f-L_n[f]\) involving the Hilbert transform. They use this to prove convergence of Lagrange interpolation for functions \(f\) on \((-1,1)\) which are the restrictions of functions \(f\) analytic in the complex plane except the two rays \((-\infty,-1]\) and \([1,\infty)\), and vanish at infinity. The convergence holds for interpolation at the zeros of certain polynomials, such as e.g. the Chebyshev and Jacobi polynomials. The hypothesis on \(f\) differs from the usual assumption, that \(f\) be analytic in an ellipse with foci at \(\pm 1\). It is satisfied, e.g., by the functions \(f(x)=(1-x^2)^{-\alpha}\) for \(0<\alpha<1\).