Introduction to additive combinatorics lecture 7.3 -- dual groups and the discrete Fourier transform

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The discrete Fourier transform is a fundamental tool in additive combinatorics that makes it possible to prove many interesting results that would be very hard or even impossible to prove otherwise. Here I discuss the characters on a finite Abelian group G, prove that they are orthogonal and form a group that is (non-naturally) isomorphic to G, known as the dual group of G, and define the Fourier transform, which takes functions defined on G to functions defined on the dual group.

0:00 Introduction
1:22 Characters on finite Abelian groups
3:22 The characters form an Abelian group under pointwise multiplication
6:02 Notation concerning inner products
7:27 Distinct characters are orthogonal
10:53 The characters span the space of functions on G
17:41 G is isomorphic to the dual group
20:39 Notation for and definitions of p-norms and inner products
25:50 Definition of the Fourier transform and discussion of two basic properties

  1. Timothy Gowers
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