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A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a cycle . [ 1 ]
This is a list of some well-known periodic functions. The constant function f ( x ) = c , where c is independent of x , is periodic with any period, but lacks a fundamental period . A definition is given for some of the following functions, though each function may have many equivalent definitions.
The Fourier transform of a periodic function cannot be defined using the integral formula directly. In order for integral in Eq.1 to be defined the function must be absolutely integrable. Instead it is common to use Fourier series. It is possible to extend the definition to include periodic functions by viewing them as tempered distributions.
It is a periodic, piecewise linear, continuous real function. Like a square wave, the triangle wave contains only odd harmonics. However, the higher harmonics roll off much faster than in a square wave (proportional to the inverse square of the harmonic number as opposed to just the inverse).
The square wave in mathematics has many definitions, which are equivalent except at the discontinuities: It can be defined as simply the sign function of a sinusoid: = () = () = () = (), which will be 1 when the sinusoid is positive, −1 when the sinusoid is negative, and 0 at the discontinuities.
Periodic functions can be identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group, denoted by or . The Fourier transform is also part of Fourier analysis , but is defined for functions on R n {\displaystyle \mathbb {R} ^{n}} .
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency.The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals.
The space B p of Besicovitch almost periodic functions (for p ≥ 1) contains the space W p of Weyl almost periodic functions. If one quotients out a subspace of "null" functions, it can be identified with the space of L p functions on the Bohr compactification of the reals.