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.

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