# Banach 27

This lecture (Banach27) was given on 10/28/09

### Excerpts

Defined the Fourier coeff. $\widehat{\mu}(n)$ associated to a finite complex Borel measure $\mu$ on $\mathbb{T}$.  The goal is to characterize $\widehat{\mu}(n)$ for such measures.  Recall an atom (in measure theory) is a set of positive measure that has not proper subset of positive measure.

Definitions

1. Fourier coeff. of $\mu$.
2. positive definite and positive semidefinite sequences $(\alpha_n)_{n \in \mathbb{Z}}$ (This can be interpreted by looking at the principal minors of an infinite matrix)

Results

1. Thm III.3.3 The seq $(\alpha_n)_{n \in \mathbb{Z}}$ is the seq of Fourier coeff. of a finite positive Borel measure $\mu$ on $\mathbb{T}$ iff it is positive semidefinite.