Dirichlet conditions

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In mathematics, the Dirichlet conditions are sufficient conditions for a real-valued, periodic function f to be equal to the sum of its Fourier series at each point where f is continuous. Moreover, the behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the discontinuity). These conditions are named after Peter Gustav Lejeune Dirichlet.

The conditions are:^[1]

1. f must be absolutely integrable over a period.
2. f must be of bounded variation in any given bounded interval.
3. f must have a finite number of discontinuities in any given bounded interval, and the discontinuities cannot be infinite.

Dirichlet's theorem for 1-dimensional Fourier series[]

We state Dirichlet's theorem assuming f is a periodic function of period 2π with Fourier series expansion where

${\displaystyle a_{n}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)e^{-inx}\,dx.}$

The analogous statement holds irrespective of what the period of f is, or which version of the Fourier expansion is chosen (see Fourier series).

Dirichlet's theorem: If f satisfies Dirichlet conditions, then for all x, we have that the series obtained by plugging x into the Fourier series is convergent, and is given by

${\displaystyle \sum _{n=-\infty }^{\infty }a_{n}e^{inx}={\frac {f(x^{+})+f(x^{-})}{2}},}$

where the notation

${\displaystyle f(x^{+})=\lim _{y\to x^{+}}f(y)}$

${\displaystyle f(x^{-})=\lim _{y\to x^{-}}f(y)}$

denotes the right/left limits of f.^[1]

A function satisfying Dirichlet's conditions must have right and left limits at each point of discontinuity, or else the function would need to oscillate at that point, violating the condition on maxima/minima. Note that at any point where f is continuous,

${\displaystyle {\frac {f(x^{+})+f(x^{-})}{2}}=f(x).}$

Thus Dirichlet's theorem says in particular that under the Dirichlet conditions the Fourier expansion for f converges to f(x) wherever f is continuous.

References[]

External links[]

• "Dirichlet conditions". PlanetMath.