Monotonicity and asymptotics of zeros of Sobolev type orthogonal polynomials: A general case

Abstract

We investigate the location, monotonicity, and asymptotics of the zeros of the polynomials orthogonal with respect to the Sobolev type inner product< p, q > (lambda,c.j) = integral(b)(a) p(x)q(x)mu(x) + lambda p((j))(c)q((j))(c),where mu is a positive Borel measure, lambda >= 0, j is an element of Z(+), and c is not an element of (a, b). We prove that these zeros are monotonic function of the parameter A and establish their asymptotics when either lambda converges to zero or to infinity. The precise location of the extreme zeros is also analyzed. (c) 2012 IMACS. Published by Elsevier B.V. All rights reserved.