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:
\((\text D 1)\) | $:$ | $f$ is absolutely integrable | |||||||
\((\text D 2)\) | $:$ | $f$ has a finite number of local maxima and local minima | |||||||
\((\text D 3)\) | $:$ | $f$ 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:
- $\map f x = \map \sin {\dfrac 1 {x - 1} }$
does not satisfy the Dirichlet conditions on the real interval $\openint 0 {2 \pi}$.
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