First and second kind paraorthogonal polynomials and their zeros

Abstract

Given a probability measure μ with infinite support on the unit circle ∂D = {z: |z| = 1}, we consider a sequence of paraorthogonal polynomials h_n(z,λ) vanishing at z = λ where λ ∈ ∂D is fixed. We prove that for any fixed z_0 ∉ supp(dμ) distinct from λ, we can find an explicit ρ > 0 independent of n such that either h_n or h_(n+1) (or both) has no zero inside the disk B(z_0,ρ), with the possible exception of λ. Then we introduce paraorthogonal polynomials of the second kind, denoted s_n(z,λ). We prove three results concerning s_n and h_n. First, we prove that zeros of s_n and h_n interlace. Second, for z_0 an isolated point in supp(dμ), we find an explicit radius ρ(over tilde) such that either s_n or s_(n+1) (or both) have no zeros inside B(z_0, ρ(over tilde)). Finally, we prove that for such z_0 we can find an explicit radius such that either h_n or h_(n+1) (or both) has at most one zero inside the ball B(z_0, ρ(over tilde)).