FoCM 2014 conference

Workshop C5 - Special Functions and Orthogonal Polynomials

December 18, 15:00 ~ 15:30 - Room A21

## Orthogonal and para-orthogonal polynomials on the unit circle

### A. Sri Ranga

When a nontrivial measure $\mu$ on the unit circle satisfies the symmetry $d\mu(e^{i(2\pi-\theta)}) = - d\mu(e^{i\theta})$ then the associated orthogonal polynomials on the unit circle, say $S_n$, are all real. In this case, in [3], Delsarte and Genin have shown that the two sequences of para-orthogonal polynomials $\{zS_{n}(z) + S_{n}^{\ast}(z)\}$ and $\{zS_{n}(z) - S_{n}^{\ast}(z)\}$ satisfy three term recurrence formulas and have explored some further consequences of these sequences of polynomials such as their connections to sequences of orthogonal polynomials on the interval $[-1,1]$. Even though results presented in Delsare and Genin [4] extend these partly to include any nontrivial measures on the unit circle, only recently, in [2] (and also [1]), the extension associated with the para-orthogonals polynomials $zS_{n}(z) - S_{n}^{\ast}(z)$ was studied extensively. The results given in [2], especially from the point of view of three term recurrence, provide also as a nice application a characterization for any pure points in the measure. The main objective of the present contribution is to provide some recent developments concerning the extension for the para-orthogonals polynomials $zS_{n}(z) + S_{n}^{\ast}(z)$ to cover all nontrivial measures on the unit circle.

References

[1] K. Castillo, M. S. Costa, A. Sri Ranga and D. O. Veronese, A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula, J. Approx. Theory, 184 (2014), 146-162.

[2] M.S. Costa, H.M. felix and A. Sri Ranga, Orthogonal polynomials on the unit circle and chain sequences, J. Approx. Theory, 173 (2013), 14-32.

[3] P. Delsarte and Y. Genin, The split Levinson algorithm, IEEE Trans. Acoust. Speech Signal Process, 34 (1986), 470-478.

[4] P. Delsarte and Y. Genin, The tridiagonal approach to Szeg\H{o}'s orthogonal polynomials, Toeplitz linear system, and related interpolation problems, SIAM J. Math. Anal., 19 (1988), 718-735.

Joint work with Cleonice F. Bracciali (Universidade Estadual Paulista, Brazil) and A. Swaminathan (IIT Roorkee, India).