# Definition:Dirichlet Conditions

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## Definition

Let $\alpha, \beta \in \R$ be a real numbers such that $\alpha < \beta$.

Let $\map f x$ be a real function which is defined and bounded on the interval $\openint \alpha \beta$.

The **Dirichlet conditions** on $f$ are sufficient conditions that $f$ must satisfy on $\openint \alpha \beta$ in order for:

- the Fourier series of $f$ at every $x$ in $\openint \alpha \beta$ to equal $\map f x$
- the behaviour of a Fourier series to be determined at finite discontinuities of $f$ in $\openint \alpha \beta$:

They are as follows:

\((\text D 1)\) | $:$ | $\quad f$ is absolutely integrable | ||||||

\((\text D 2)\) | $:$ | $\quad f$ has a finite number of local maxima and local minima | ||||||

\((\text D 3)\) | $:$ | $\quad f$ has a finite number of discontinuities, all of them finite |

## Examples

### Reciprocal of $4 - x^2$

The function:

- $\map f x = \dfrac 1 {4 - x^2}$

does not satisfy the Dirichlet conditions on the real interval $\openint 0 {2 \pi}$.

### Sine of $\dfrac 1 {x - 1}$

The function:

- $f \left({x}\right) = \sin \left({\dfrac 1 {x - 1} }\right)$

does not satisfy the Dirichlet conditions on the real interval $\left({0 \,.\,.\, 2 \pi}\right)$.

## Source of Name

This entry was named for Johann Peter Gustav Lejeune Dirichlet.

## Sources

- 1961: I.N. Sneddon:
*Fourier Series*... (previous) ... (next): Chapter One: $\S 2$. Fourier Series