14 Facts About Fourier analysis

Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.

Each transform used for analysis has a corresponding inverse transform that can be used for synthesis.

Fourier analysis has many scientific applications – in physics, partial differential equations, number theory, combinatorics, signal processing, digital image processing, probability theory, statistics, forensics, option pricing, cryptography, numerical analysis, acoustics, oceanography, sonar, optics, diffraction, geometry, protein structure analysis, and other areas.

Fourier analysis transformation is useful as a compact representation of a signal.

In image reconstruction, each image square is reassembled from the preserved approximate Fourier analysis-transformed components, which are then inverse-transformed to produce an approximation of the original image.

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Fourier analysis transforms are not limited to functions of time, and temporal frequencies.

When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate narrowband components of a compound waveform, concentrating them for easier detection or removal.

Inverse transform, known as Fourier analysis series, is a representation of in terms of a summation of a potentially infinite number of harmonically related sinusoids or complex exponential functions, each with an amplitude and phase specified by one of the coefficients:.

Similarly, finite-duration functions can be represented as a Fourier analysis series, with no actual loss of information except that the periodicity of the inverse transform is a mere artifact.

In modern times, variants of the discrete Fourier analysis transform were used by Alexis Clairaut in 1754 to compute an orbit, which has been described as the first formula for the DFT, and in 1759 by Joseph Louis Lagrange, in computing the coefficients of a trigonometric series for a vibrating string.

An early modern development toward Fourier analysis was the 1770 paper Reflexions sur la resolution algebrique des equations by Lagrange, which in the method of Lagrange resolvents used a complex Fourier decomposition to study the solution of a cubic:Lagrange transformed the roots into the resolvents:.

Fourier analysis'storians are divided as to how much to credit Lagrange and others for the development of Fourier theory: Daniel Bernoulli and Leonhard Euler had introduced trigonometric representations of functions, and Lagrange had given the Fourier series solution to the wave equation, so Fourier's contribution was mainly the bold claim that an arbitrary function could be represented by a Fourier series.

Subsequent development of the field is known as harmonic Fourier analysis, and is an early instance of representation theory.

In signal processing terms, a function is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier analysis transform has perfect frequency resolution, but no time information.