Orthogonal Polynomials and Krein’s Theorem

We start this chapter with the study of the problem of orthogonalizing the power functions 0.1$$ 1,z,z^2 ,... $$ on the unit circle T with respect to a positive definite scalar product on L₂(T) given by 0.2$$ \left\langle {f,g} \right\rangle _\omega = \frac{1} {{2\pi }}\int_{ - \pi }^\pi {\overline {g(e^{it} )} } \omega (e^{it} )f(e^{it} )dt $$ Here ω is a positive integrable function. The functions that result from orthogonalizing the functions in (0.1) for the scalar product in (0.2) are polynomials, called the Szegő polynomials corresponding to the weight function ω. Szegő ‘s proved that these polynomials have all their zeros in the open unit disk. This result, which we refer to as Szegő ‘s Theorem, is proved in Section 1.1.