Szegő polynomial explained

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

\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

\phi _{n(z)} for the polynomials, they obey a recurrence relation\phi _{n+1} (z) = z\phi _{n(z)} + \rho _{n+1} \phi ^{*(z)} _{n}

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

• • G. Szegő, "Orthogonal polynomials", Colloq. Publ., 33, Amer. Math. Soc. (1967)