Fourier series: Difference between revisions

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==The Fourier Series==
==The Fourier Series==
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.
<center>

<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>.
</center>







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