Szegő polynomial

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

${\displaystyle \langle f|g\rangle =\int _{-\pi }^{\pi }f(e^{i\theta }){\overline {g(e^{i\theta })}}\,d\mu }$

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

${\displaystyle \phi _{n+1}(z)=z\phi _{n}(z)+\rho _{n+1}\phi _{n}^{*}(z)}$

where ${\displaystyle \rho _{n+1}}$ is a parameter, called the reflection coefficient or the Szegő parameter.

See also

• Cayley transform
• Schur class
• Favard's theorem

References

• Bultheel, A. (2001) [1994], "Szegö polynomial", Encyclopedia of Mathematics, EMS Press
• G. Szegő, "Orthogonal polynomials", Colloq. Publ., 33, Amer. Math. Soc. (1967)