# How far is 1 from the subspace spanned by polynomials vanishing at the origin?

Speaker: Md. Ramiz Reza

Date: 21th April, 2015 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

I will talk about Szego’s Theorem. The set up for the theorem is as follows. Let $\mu$ be a positive measure on the unit circle and we want to know in the Hilbert space $L^2(\mu)$ what the distance from the constant function 1 is to the subspace spanned by polynomials which vanish at the origin. Szego’s theorem states that the square of the distance (infimum) above is equal to $\exp \{ \frac{1}{2\pi} \int \limits _{0}^{2\pi} \log h d\theta \}$, where $h$ is the derivative of $\mu$ w.r.t the normalized Lebesgue measure.

In this process we will also prove the famous theorem of F and M Riesz which tells us that if a complex Borel measure on the unit circle is analytic, i.e., $\int \exp(i n \theta) d\mu(\theta) = 0$ for all $n\geq 1$, then $\mu$ is absolutely continuous w.r.t Lebesgue measure.

Area: Functional Analysis