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.
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