an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines
a (possibly infinite) sum of cosine waveforms with the property that every frequency is a positive integer multiple of a fundamental frequency,
the spectrum of a periodic waveform, which is composed of discrete harmonics. See also Fourier transform.
an infinite series in which the terms are constants multiplied by sine or cosine functions of integer multiples of the variable and which is used in the analysis of periodic functions
A series proposed by the French mathematician Fourier about the year 1807. The series involves the sines and cosines of whole multiples of a varying angle and is usually written in the following form: y = Ho + A1 sin x + A2 sin 2x + A3 sin 3x + ... B1 cos x + B2 cos 2x + B3 cos 3x + ... By taking a sufficient number of terms the series may be made to represent any periodic function of x.
A mathematical series that shows any periodic function to be a combination of sine and cosine terms.
The representation of a function () in an interval (−, ) by a series consisting of sines and cosines with a common period, in the form where the Fourier coefficients are defined as and When () is an even function, only the cosine terms appear; when () is odd, only the sine terms appear. The conditions on () 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/ ( = 1, 2, . . .); the Fourier integral requires a continuous spectrum. See also Fourier transform.
Application of the Fourier theorem to a periodic function, resulting in sine and cosine terms which are harmonics of the periodic frequency. [After Baron Jean Baptiste Joseph Fourier.