Editing Periodic function (section)

===Quotient spaces as domain===
In [[signal processing]] you encounter the problem, that [[Fourier series]] represent periodic functions and that Fourier series satisfy [[convolution theorem]]s (i.e. [[convolution]] of Fourier series corresponds to multiplication of represented periodic function and vice versa), but periodic functions cannot be convolved with the usual definition, since the involved integrals diverge. A possible way out is to define a periodic function on a bounded but periodic domain. To this end you can use the notion of a [[quotient space (linear algebra)|quotient space]]:

:<math>{\mathbb{R}/\mathbb{Z}}
 = \{x+\mathbb{Z} : x\in\mathbb{R}\}
 = \{\{y : y\in\mathbb{R}\land y-x\in\mathbb{Z}\} : x\in\mathbb{R}\}</math>.

That is, each element in <math>{\mathbb{R}/\mathbb{Z}}</math> is an [[equivalence class]] of [[real number]]s that share the same [[fractional part]]. Thus a function like <math>f : {\mathbb{R}/\mathbb{Z}}\to\mathbb{R}</math> is a representation of a 1-periodic function.