Szegő polynomial

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

$⟨f|g⟩=\underset{-\pi }{\overset{\pi }{\int }}f(e^{i\theta })\overline{g(e^{i\theta })}d\mu$

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

${\varphi }_{n+1}(z)=z{\varphi }_n(z)+{\rho }_{n+1}{\varphi }_n^{*}(z)$

where ${\rho }_{n+1}$ 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)