# Mathematics MA342A

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JS & SS Mathematics, and

JS & SS TSM Mathematics students

### Module MA342A, Harmonic Analysis I: Richard M. Timoney

## Introduction

A key goal in this module is to introduce Fourier analysis in both classical settings (periodic functions on intervals and functions on the real line) and abstract contexts of locally compact abelian groups.

Fourier had the idea that all functions could be reconstructed as superpositions of sines and cosines (with various periods). From our point of view, it is more convenient to work with complex valued functions and complex exponentials instead of sines and cosines. This approach lends itself to a generalization that unifies Fourier analysis across different contexts.

We will begin with the (probably familiar) definition of the Fourier series of a periodic function and give examples for the other contexts of locally compact abelian groups. This will allow an introduction which is relatively easy to follow, but at the expense of leaving most significant results to later.

There is a sample exam for this module here.

See the Syllabus for more specific information.