Fourier Series

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Fourier Series </dt>
The representation of a function f(x) in an interval (-L, L) by a series consisting of sines and cosines with a common period 2L, in the form,

Missing Image:img src="SP7-f_files/113Fser1.html" height="74" width="304"where the

Fourier coefficients are defined as

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</p>Missing Image:img src="SP7-f_files/113Fser3.html" height="38" width="192"

and

</p>Missing Image:img src="SP7-f_files/113Fser4.html" height="37" width="188"

</p> </dd>
'When f(x) is an even function, only the cosine terms appear; when f(x) is odd, only the sine terms appear. '</dd>
'The conditions on f(x) guaranteeing convergence of the series are quite general, and the series may serve as a root-mean-square approximation even when it does not converge. '</dd>
'If the function is defined on an infinite interval and is not periodic, it is represented by the Fourier integral. By either representation, the function is decomposed into periodic components whose frequencies constitute the spectrum of the function. The Fourier series employs a discrete spectrum of wavelengths 2 L / n(n = 1,2,...); the Fourier integral requires a continuous spectrum. '</dd>
'See Fourier transform. [[/a>|/a> ]]</dd>

References

This article is based on NASA's Dictionary of Technical Terms for Aerospace Use