Fourier series: Difference between revisions
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<div class="definition"><div class="short_definition">The representation of a function ''f''(''x'') in an interval ( | <div class="definition"><div class="short_definition">The representation of a function ''f''(''x'') in an interval (-''L'', ''L'') by a series consisting of sines and cosines with a common [[period]] 2''L'', in the form <div class="display-formula"><blockquote>[[File:ams2001glos-Fe10.gif|link=|center|ams2001glos-Fe10]]</blockquote></div> where the [[Fourier coefficients]] are defined as <div class="display-formula"><blockquote>[[File:ams2001glos-Fe11.gif|link=|center|ams2001glos-Fe11]]</blockquote></div> <div class="display-formula"><blockquote>[[File:ams2001glos-Fe12.gif|link=|center|ams2001glos-Fe12]]</blockquote></div> and <div class="display-formula"><blockquote>[[File:ams2001glos-Fe13.gif|link=|center|ams2001glos-Fe13]]</blockquote></div></div><br/><div class="paragraph">When ''f''(''x'') is an even function, only the cosine terms appear; when ''f''(''x'') is odd, only the sine terms appear. 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. 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 the frequencies of which 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]]. <br/>''See also'' [[Fourier transform]].</div><br/> </div> | ||
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Revision as of 15:15, 20 February 2012
Fourier series
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 where the Fourier coefficients are defined as and
When f(x) is an even function, only the cosine terms appear; when f(x) is odd, only the sine terms appear. 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. 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 the frequencies of which constitute the spectrum of the function. The Fourier series employs a discrete spectrum of wavelengths 2L/n (n = 1, 2, . . .); the Fourier integral requires a continuous spectrum.
See also Fourier transform.
See also Fourier transform.