First and second kind paraorthogonal polynomials and their zeros

Abstract

Given a probability measure with infinite support on the unit circle *D = {z : |z| = 1}, we consider a sequence of paraorthogonal polynomials h n (z, ) vanishing at z = where ∈ *D is fixed. We prove that for any fixed z 0 / ∈ supp(d ) distinct from , we can find an explicit > 0 independent of n such that either h n or h n+1 (or both) has no zero inside the disk B(z 0 , ), with the possible exception of .Then we introduce paraorthogonal polynomials of the second kind, denoted s n (z, ). 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(d ), we find an explicit radius˜ such that either s n or s n+1 (or both) have no zeros inside B(z 0 ,˜ ). 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 ,˜ ).

Untitled

“…The interlacing property on ‫ބ∂‬ of the zeros of {ϕ n } n≥0 was recently proved in [Dimitrov and Sri Ranga 2013, Theorem 1.1]; see also [Castillo et al 2014], although it was first proved in [Delsarte and Genin 1988, Section 5]. In any case, an easy computation shows that these polynomials satisfy the conditions of Theorem 3.4 with all their zeros on ‫.ބ∂‬ So, the interlacing property of the zeros of {ϕ n } n≥0 is also a direct consequence of the fact that from Theorem 3.4, these polynomials are POPUC associated with some nontrivial probability measure dσ supported on ‫.ބ∂‬ Actually, we can say much more about the behavior of their zeros using the known results for the zeros of POPUC; see, e.g., [Cantero et al 2002;Golinskii 2002;Simon 2005a;2005b;Wong 2007].…”

“…The results in this direction can be divided into two sets, depending on the methodology used by the authors. The first one is composed by Cantero, Moral and Velázquez [Cantero et al 2002], Golinskii [2002], and Wong [2007], whose basic tool is the Christoffel-Darboux formula. The second one is by Simon [2007] who used the theory of rank-one perturbations of unitary matrices.…”

“…properties [Cantero et al 2002;Golinskii 2002;Simon 2005a;2005b;Wong 2007]. These polynomials play in the unit circle the same role as orthogonal polynomials on the real line (OPRL) from the perspective of quadrature formulas.…”

“…The uses of their zeros instead of zeros of OPUC in frequency analysis problems [Daruis et al 2003] and their appearance as the minimizer of the isometric Arnoldi minimization problem [Helsen et al 2005] represent some of their best applications. On the other hand, the works of Cantero, Moral and Velázquez [Cantero et al 2002], Golinskii [2002], Simon [2007], and Wong [2007] are essential to understand the behavior of the zeros of POPUC. Recently in [Simanek 2015], it was also proved that the zeros of invariant POPUC designate the location of a set of particles that are in electrostatic equilibrium with respect to a particular external field.…”

Orthogonal Polynomials Associated with Complementary Chain Sequences

“…, n − 1, X n , 1 = 0, X n , z n = 0, which are termed as deficiency in the orthogonality of these para-orthogonal polynomials. In recent years, these para-orthogonal polynomials have been linked to kernel polynomials K n (z, ω), see [6,11,27]. The kernel polynomials K n (z, ω) satisfy the Christoffel-Darboux formula…”