Dirichlet conditions

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

## Readings

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* Wikipedia: Dirichlet conditions

## Conditions

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### Condition 1

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

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

### Condition 2

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

### Condition 3

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