Fourier analysis

n.
The branch of mathematics concerned with the approximation of periodic functions by the Fourier series and with generalizations of such approximations to a wider class of functions.

The representation of a function f(x), which is periodic in x, as an infinite series of sine and cosine functions: A series of this type is called a Fourier series. If the function is periodic with a period 2π, the coefficients a₀, a_n, b_n are: Fourier analysis and Fourier series are named after the French mathematician and engineer Joseph Fourier (1768–1830). Fourier series have many important applications in mathematics, science, and engineering, having been invented by Fourier in the first quarter of the 19th century in his analysis of the problem of heat conduction.

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A type of mathematical analysis that attempts to identify patterns or cycles in a time series data set which has already been normalized. By first removing any effects of trends or other complicating factors from the data set, the effects of periodic cycles or patterns can be identified more accurately, leaving the analyst with a good estimate of the direction that the data under analysis will take in the future. Named after the nineteenth-century French mathematician and physicist Joseph Fourier.

Investopedia Says:
This type of analysis may sound complex, but it actually makes good sense. For example, suppose a manufacturing company wanted to know what stage of its price cycle its main raw material was in. Because inflation would constantly be increasing the dollar price of the commodity over time, an analyst would remove the effects of inflation from the commodity's historical prices first. Once inflation was controlled for, the analyst would then have a much more accurate picture of the price cycles experienced by the commodity.

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A method of analysing into simpler components any complex sound or other waveform (whether it describes fluctuations over time or over place). The analysis uses the Fourier theorem, which states that any function or waveform can be described as a series of sine waves of multiple frequencies and various amplitudes. Conversely, the characteristic timbre of musical instruments can be produced by combining separate sine wave oscillations (Fourier synthesis).

Fourier analysis is useful for studies of electroencephalography and speech recognition by machine. The notion that visual pattern recognition may employ Fourier analysis is due largely to the work of F. W. Campbell at the University of Cambridge.

The Fourier theorem was formulated in 1826 by the French mathematician and physicist Jean Baptiste Joseph de Fourier (1768–1830), who was a professor at the École Polytechnique in Paris from 1795 to 1798; he accompanied Napoleon to Egypt as scientific adviser and was for a time governor of Lower Egypt. He was created baron in 1808 and in 1827 became president of the council of the École Polytechnique.

(Published 1987)

Bibliography
• Campbell, F. W., and Robson, J. G. (1968). 'Application of Fourier analysis to the visibility of gratings'. Journal of Physiology, 197.
• Herivel, J. (1975). Joseph Fourier: The Man and the Physicist.

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