Dirichlet conditions

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

Readings

Wikipedia: Dirichlet conditions

Conditions

Condition 1

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

$\int_T|x(t)|<\infty$

Condition 2

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

Condition 3

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

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