A Hilbert transform representation of the error in Lagrange interpolation

Let Ln [f ] denote the Lagrange interpolation polynomial to a function f at the zeros of a polynomial Pn with distinct real zeros. We show that f − Ln [f ] = −PnHe [ H [f ] Pn ] , where H denotes the Hilbert transform, and He is an extension of it. We use this to prove convergence of Lagrange interpolation for certain functions analytic in (−1, 1) that are not assumed analytic in any ellipse with foci at (−1, 1).