Editing Periodic function (section)

==Properties==
<!-- '''periodicity with period zero''' ''P'' ''' greater than zero if !-->
Periodic functions can take on values many times. More specifically, if a function <math>f</math> is periodic with period <math>P</math>, then for all <math>x</math> in the domain of <math>f</math> and all positive integers <math>n</math>,

: <math>f(x + nP) = f(x)</math>

If <math>f(x)</math> is a function with period <math>P</math>, then <math>f(ax)</math>, where <math>a</math> is a non-zero real number such that <math>ax</math> is within the domain of <math>f</math>, is periodic with period <math display="inline">\frac{P}{a}</math>. For example, <math>f(x) = \sin(x)</math> has period <math>2 \pi</math> and, therefore, <math>\sin(5x)</math> will have period <math display="inline">\frac{2\pi}{5}</math>.

Some periodic functions can be described by [[Fourier series]]. For instance, for [[Lp space|''L''<sup>2</sup> functions]], [[Carleson's theorem]] states that they have a [[pointwise]] ([[Lebesgue measure|Lebesgue]]) [[almost everywhere convergence|almost everywhere convergent]] [[Fourier series]]. Fourier series can only be used for periodic functions, or for functions on a bounded (compact) interval. If <math>f</math> is a periodic function with period <math>P</math> that can be described by a Fourier series, the coefficients of the series can be described by an integral over an interval of length <math>P</math>.

Any function that consists only of periodic functions with the same period is also periodic (with period equal or smaller), including:
* addition, [[subtraction]], multiplication and division of periodic functions, and
* taking a power or a root of a periodic function (provided it is defined for all <math>x</math>).