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

Research paper by C.F. Bracciali, A. Sri Ranga, A. Swaminathan

Indexed on: 14 Jun '16Published on: 11 Jun '16Published in: Applied Numerical Mathematics



Abstract

When a nontrivial measure μ   on the unit circle satisfies the symmetry dμ(ei(2π−θ))=−dμ(eiθ)dμ(ei(2π−θ))=−dμ(eiθ) then the associated orthogonal polynomials on the unit circle, say ΦnΦn, are all real. In this case, in 1986, Delsarte and Genin have shown that the two sequences of para-orthogonal polynomials <img height="15" border="0" style="vertical-align:bottom" width="104" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0168927416300964-si3.gif">{zΦn(z)+Φn⁎(z)} and <img height="15" border="0" style="vertical-align:bottom" width="104" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0168927416300964-si4.gif">{zΦn(z)−Φn⁎(z)}, where <img height="18" border="0" style="vertical-align:bottom" width="116" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0168927416300964-si5.gif">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][−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 <img height="15" border="0" style="vertical-align:bottom" width="95" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0168927416300964-si7.gif">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 <img height="15" border="0" style="vertical-align:bottom" width="95" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0168927416300964-si8.gif">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 <img height="21" border="0" style="vertical-align:bottom" width="142" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0168927416300964-si9.gif">∫02π|eiθ−w|−2dμ(eiθ), where w   is such that |w|=1|w|=1, is given in terms of the coefficients <img height="16" border="0" style="vertical-align:bottom" width="96" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0168927416300964-si11.gif">αn−1=−Φn(0)‾, n≥1n≥1. Examples are also provided to justify all the results.