Editing Periodic function (section)

==Generalizations==

===Antiperiodic functions===
One subset of periodic functions is that of '''antiperiodic functions'''. This is a function <math>f</math> such that <math>f(x+P) = -f(x)</math> for all <math> x</math>. For example, the sine and cosine functions are <math>\pi</math>-antiperiodic and <math>2\pi</math>-periodic. While a <math> P</math>-antiperiodic function is a <math> 2P</math>-periodic function, the [[converse (logic)|converse]] is not necessarily true.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Antiperiodic Function |url=https://mathworld.wolfram.com/ |access-date=2024-06-06 |website=mathworld.wolfram.com |language=en}}</ref>

===Bloch-periodic functions===
A further generalization appears in the context of [[Bloch's theorem]]s and [[Floquet theory]], which govern the solution of various periodic differential equations.  In this context, the solution (in one dimension) is typically a function of the form
:<math>f(x+P) = e^{ikP} f(x) ~,</math>
where <math>k</math> is a real or complex number (the ''Bloch wavevector'' or ''Floquet exponent'').  Functions of this form are sometimes called '''Bloch-periodic''' in this context.   A periodic function is the special case <math>k=0</math>, and an antiperiodic function is the special case <math>k=\pi/P</math>. Whenever <math>k P/ \pi</math> is rational,  the function is also periodic.

===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.