Nonnegative Quadratic Forms and Bounds on Orthogonal Polynomials

Abstract

We show that some nonnegative quadratic forms containing orthogonal polynomials, such as e.g. the Christoffel-Darboux kernel for x=y in the classical case, provide a lot of information about behavior of the polynomials on the real axis. We illustrate the method for the case of Hermite polynomials and use it to derive new explicit bounds for binary Krawtchouk polynomials. © 2001 Academic Press.