Editing Discrete Fourier transform (section)

=== Representation theory ===
{{further|Representation theory of finite groups#Discrete Fourier transform}}

The DFT can be interpreted as a complex-valued [[representation theory|representation]] of the finite [[cyclic group]]. In other words, a sequence of <math>n</math> complex numbers can be thought of as an element of <math>n</math>-dimensional complex space <math>\mathbb{C}^n</math> or equivalently a function <math>f</math> from the finite cyclic group of order <math>n</math> to the complex numbers, <math>\mathbb{Z}_n \mapsto \mathbb{C}</math>. So  <math>f</math> is a [[class function]] on the finite cyclic group, and thus can be expressed as a linear combination of the irreducible characters of this group, which are the roots of unity.

From this point of view, one may generalize the DFT to representation theory generally, or more narrowly to the [[representation theory of finite groups]].

More narrowly still, one may generalize the DFT by either changing the target (taking values in a field other than the complex numbers), or the domain (a group other than a finite cyclic group), as detailed in the sequel.