Para-orthogonal polynomials on the unit circle satisfying three term recurrence formulas

Abstract

When a nontrivial measure μ on the unit circle satisfies the symmetry dμ(ei(2π−θ))=−dμ(eiθ) then the associated orthogonal polynomials on the unit circle, say Φn, are all real. In this case, in 1986, Delsarte and Genin have shown that the two sequences of para-orthogonal polynomials {zΦn(z)+Φn⁎(z)} and {zΦn(z)−Φn⁎(z)}, where Φn⁎(z)=znΦn(1/z‾)‾, satisfy three term recurrence formulas and have also explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval [−1,1]. The same authors, in 1988, have also provided a means to extend these results to cover any nontrivial measure on the unit circle. However, only recently the extension associated with the para-orthogonal polynomials zΦn(z)−Φn⁎(z) was thoroughly explored, especially from the point of view of three term recurrence and chain sequences. The main objective of the present article is to provide the theory surrounding the extension associated with the para-orthogonal polynomials zΦn(z)+Φn⁎(z) for any nontrivial measure on the unit circle. As an important application of the theory, a characterization for the existence of the integral ∫02π|eiθ−w|−2dμ(eiθ), where w is such that |w|=1, is given in terms of the coefficients αn−1=−Φn(0)‾, n≥1. Examples are also provided to justify all the results.