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

摘要

We investigate the location, monotonicity, and asymptotics of the zeros of the polynomials orthogonal with respect to the Sobolev type inner product〈p,q〉λ.c.j = b∫a p(x)q(x)dμ(x) + λp~((j))(c)q~((j))(c), where μ is a positive Borel measure, λ≥ 0, j ∈ Z_+, and c € (a, b). We prove that these zeros are monotonic function of the parameter λ and establish their asymptotics when either A converges to zero or to infinity. The precise location of the extreme zeros is also analyzed.