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| {{distinguish|Dirichlet boundary condition}}
| | Greetings! I am Myrtle Shroyer. Bookkeeping is what I do. Minnesota is exactly where he's been residing for many years. One of the very very best issues in the globe for me is to do aerobics and I've been doing it for quite a while.<br><br>Have a look at my blog post ... [http://smoothbuss.herobo.com/index.php?do=/profile-10614/info/ home std test kit] |
| In [[mathematics]], the '''Dirichlet conditions''' are [[sufficient condition]]s for a [[real numbers|real]]-valued, [[periodic function]] ''f''(''x'') to be equal to the sum of its [[Fourier series]] at each point where ''f'' is [[continuous function|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]].
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| The conditions are:
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| *''f''(''x'') must be [[absolutely integrable]] over a period.
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| *''f''(''x'') must have a finite number of [[Maxima_and_minima|extrema]] in any given interval, i.e. there must be a finite number of maxima and minima in the interval.
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| *''f''(''x'') must have a finite number of [[Classification_of_discontinuities|discontinuities]] in any given interval, however the discontinuity cannot be infinite.
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| *''f''(''x'') must be [[bounded function|bounded]]
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| The last three conditions are satisfied if ''f'' is a function of [[bounded variation]] over a period.
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| ==Dirichlet's Theorem for 1-Dimensional Fourier Series==
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| We state Dirichlet's theorem assuming ''f'' is a periodic function of period 2π with Fourier series expansion where
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| :<math> a_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx}\, dx. </math>
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| The analogous statement holds irrespective of what the period of ''f'' is, or which version of the Fourier expansion is chosen (see [[Fourier series]]).
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| :'''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 | |
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| ::<math> \sum_{n = -\infty}^\infty a_n e^{inx} = \frac{1}{2}(f(x+) + f(x-)) </math>,
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| :where the notation
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| ::<math> f(x+) = \lim_{y \to x^+} f(y) </math>
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| ::<math> f(x-) = \lim_{y \to x^-} f(y) </math>
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| :denotes the right/left limits of ''f''.
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| <br>
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| 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,
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| :<math> \frac{1}{2}(f(x+) + f(x-)) = f(x) </math>.
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| Thus Dirichlet's theorem says in particular that under the Dirichlet conditions the Fourier series for ''f'' converges and is equal to ''f'' wherever ''f'' is continuous.
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| ==External links==
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| *{{planetmath reference|id=3891|title=Dirichlet conditions}}
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| [[Category:Fourier series]]
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| [[Category:Theorems in analysis]]
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Greetings! I am Myrtle Shroyer. Bookkeeping is what I do. Minnesota is exactly where he's been residing for many years. One of the very very best issues in the globe for me is to do aerobics and I've been doing it for quite a while.
Have a look at my blog post ... home std test kit