Szegő polynomial

In mathematics, a Szegő polynomial is one of a family of orthogonal polynomials for the Hermitian inner product

[math]\displaystyle{ \langle f|g\rangle = \int_{-\pi}^{\pi}f(e^{i\theta})\overline{g(e^{i\theta})}\,d\mu }[/math]

where dμ is a given positive measure on [−π, π]. Writing [math]\displaystyle{ \phi_n(z) }[/math] for the polynomials, they obey a recurrence relation

[math]\displaystyle{ \phi_{n+1}(z)=z\phi_n(z) + \rho_{n+1}\phi_n^*(z) }[/math]

where [math]\displaystyle{ \rho_{n+1} }[/math] is a parameter, called the reflection coefficient or the Szegő parameter.

See also

• Cayley transform
• Schur class
• Favard's theorem

References

• Hazewinkel, Michiel, ed. (2001), "Szegö polynomial", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=s/s130650 
• G. Szegő, "Orthogonal polynomials", Colloq. Publ., 33, Amer. Math. Soc. (1967)