Dirichlet conditions

Dirichlet conditions guarantee that a periodic function ${\displaystyle x(t)}$ can be exactly represented by its Fourier transform.

Conditions

Condition 1

The function must be absolutely integrable over a single period ${\displaystyle T}$ . This is equivalent to the statement that the area enclosed between the abcissa and the function is finite over a single period.

${\displaystyle \int _{T}|x(t)|<\infty }$

Condition 2

Given any finite period of time the number of local maxima and minima of ${\displaystyle x(t)}$  within that period is finite.

Condition 3

Given any finite period of time there is a finite number of discontinuities in the function ${\displaystyle x(t)}$