A Hilbert transform representation of the error in Lagrange interpolation

AbstractLet Ln[f] denote the Lagrange interpolation polynomial to a function f at the zeros of a polynomial Pn with distinct real zeros. We show thatf−Ln[f]=−PnHeH[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)