Higher order Fourier analysis

Traditionally, Fourier analysis has been focused the analysis of func- tions in terms of linear phase functions such as the sequence n ! e(n) = e2in. In recent years, though, applications have arisen - particularly in connection with problems involving linear patterns such as arithmetic progressions - in which it has been necessary to go beyond the linear phases, replacing them to higher order functions such as quadratic phases n ! e(n2). This has given rise to the sub- ject of quadratic Fourier analysis, and more generally to higher order Fourier analysis.