Interpolation beyond the interval of convergence: An extension of Erdos-Turan Theorem

摘要

An elegant result due to Erdos and Turan states that the sequence of Lagrange interpolants to a continuous function f at the zeros of orthogonal polynomi-als over an interval [c, d] converges to f in mean square. We introduce certain sequences of polynomials which preserve both interpolation as well as conver-gence properties of Erdos-Turan Theorem. In addition, they interpolate f at a finite number of pre-assigned points lying outside [c, d]. We shall introduce a method to construct the suggested polynomials and also investigate their properties. Some computational aspects are also discussed.