**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 be a positive measure on the unit circle and we want to know in the Hilbert space 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 , where is the derivative of 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., for all , then is absolutely continuous w.r.t Lebesgue measure.

**Area**: Functional Analysis