Fourier series: Difference between revisions
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==The Fourier Series== |
==The Fourier Series== |
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A Fourier series is an expansion of a periodic function <math>f</math> in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. |
A Fourier series is an expansion of a periodic function <math>f</math> in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. |
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<math> f(t) = \sum_{k= \infty}^ \infty \alpha_k e^ \frac{j 2 \pi k t}{T} </math>. |
<math> f(t) = \sum_{k= \infty}^ \infty \alpha_k e^ \frac{j 2 \pi k t}{T} </math>. |
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</center> |
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Revision as of 11:45, 28 October 2004
Diriclet Conditions
The conditions for a periodic function with period 2L to have a convergent Fourier series.
Theorem:
Let be a piecewise regular real-valued function defined on some interval [-L,L], such that has only a finite number of discontinuities and extrema in [-L,L]. Then the Fourier series of this function converges to when is continuous and to the arithmetic mean of the left-handed and right-handed limit of at a point where it is discontinuous.
The Fourier Series
A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions.
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see also:Orthogonal Functions
Principle author of this page: Aric Goe