Huygens ' principle when applied to an aperture simply says that the far-field diffraction pattern is the spatial Fourier transform of the aperture shape, and this is a direct by-product of using the parallel-rays approximation, which is identical to doing a plane wave decomposition of the aperture plane fields ( see Fourier optics ).

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

* Fourier optics

All of the results from geometrical optics can be recovered using the techniques of Fourier optics which apply many of the same mathematical and analytical techniques used in acoustic engineering and signal processing.

The term Fourier transform spectroscopy reflects the fact that in all these techniques, a Fourier transform is required to turn the raw data into the actual spectrum, and in many of the cases in optics involving interferometers, is based on the Wiener – Khinchin theorem.

The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, thin-walled shell theory, etc.

Fourier optics is the study of classical optics using Fourier transforms and can be seen as the dual of the Huygens-Fresnel principle.

In Fourier optics, by contrast, the wave is regarded as a superposition of plane waves which are not related to any identifiable sources ; instead they are the natural modes of the propagation medium itself.

Fourier optics forms much of the theory behind image processing techniques, as well as finding applications where information needs to be extracted from optical sources such as in quantum optics.

To put it in a slightly more complex way, similar to the concept of frequency and time used in traditional Fourier transform theory, Fourier optics makes use of the spatial frequency domain ( k < sub > x </ sub >, k < sub > y </ sub >) as the conjugate of the spatial ( x, y ) domain.

Fourier optics begins with the homogeneous, scalar wave equation ( valid in source-free regions ):

Fourier optics is somewhat different from ordinary ray optics typically used in the analysis and design of focused imaging systems such as cameras, telescopes and microscopes.

Unfortunately, ray optics does not explain the operation of Fourier optical systems, which are in general not focused systems.

This more general wave optics accurately explains the operation of Fourier optics devices.

These uniform plane waves form the basis for understanding Fourier optics.

If the Greek letter is used, it is assumed to be a Fourier transform of another function,

Also, mass spectrometry is categorized by approaches of mass analyzers: magnetic-sector, quadrupole mass analyzer, quadrupole ion trap, time-of-flight, Fourier transform ion cyclotron resonance, and so on.

* The Small-angle X-ray scattering intensity of a nanostructured system is the Fourier transform of the spatial autocorrelation function of the electron density.

* Autocorrelation in space rather than time, via the Patterson function, is used by X-ray diffractionists to help recover the " Fourier phase information " on atom positions not available through diffraction alone.

which is used to expand a plane wave as a sum of cylindrical waves, or to find the Fourier series of a tone-modulated FM signal.

This formula is useful especially when working with Fourier transforms.

This orthogonality relation can then be used to extract the coefficients in the Fourier – Bessel series, where a function is expanded in the basis of the functions J < sub > α </ sub >( x u < sub > α, m </ sub >) for fixed α and varying m.

which is just the Fourier transform of the probability density.

Although computing a power spectrum from a map is in principle a simple Fourier transform, decomposing the map of the sky into spherical harmonics, in practice it is hard to take the effects of noise and foreground sources into account.

A sequence of samples from a measuring device produces a time or spatial domain representation, whereas a discrete Fourier transform produces the frequency domain information, that is the frequency spectrum.

Often the Fourier transform is converted to the power spectrum, which is the magnitude of each frequency component squared.

As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information ( location in time ).

The lower half is its 2D Fourier transform approximately reconstructing the shape of the aperture.

In mathematics, the discrete Fourier transform ( DFT ) is a specific kind of discrete transform, used in Fourier analysis.

Therefore it is often said that the DFT is a transform for Fourier analysis of finite-domain discrete-time functions.

Like the Fourier transform, the Laplace transform is used for solving differential and integral equations.

For more details on the sequences of operations used for various quantum algorithms, see universal quantum computer, Shor's algorithm, Grover's algorithm, Deutsch-Jozsa algorithm, amplitude amplification, quantum Fourier transform, quantum gate, quantum adiabatic algorithm and quantum error correction.

Auguste Comte used the term " science social " to describe the field, taken from the ideas of Charles Fourier ; Comte also referred to the field as social physics.

To do this, X-ray scattering is used to collect data about its Fourier transform F ( q ), which is inverted mathematically to obtain the density defined in real space, using the formula

The corresponding formula for a Fourier transform will be used below

At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse.

In spectroscopy the term is used in a related sense to indicate that the experiment is performed with a mixture of frequencies at once and their respective response unravelled afterwards using the Fourier transform principle.

The method of Fourier transform spectroscopy can also be used for absorption spectroscopy.

Accordingly, the technique of " Fourier transform spectroscopy " can be used both for measuring emission spectra ( for example, the emission spectrum of a star ), and absorption spectra ( for example, the absorption spectrum of a liquid ).

Thus Fourier analysis might be used to decompose a sound into a unique combination of pure tones of various pitches.

He was the first to realize that the concentric spheres of Eudoxus of Cnidus and Callippus, unlike those used by many astronomers of later times, were not to be taken as material objects, but only as part of an algorithm similar to the modern Fourier series.

* Short-time Fourier transform or short-term Fourier transform ( STFT ), a Fourier transform during a short term of time, used in the area of signal analysis

* Fractional Fourier transform ( FRFT ), a linear transformation generalizing the Fourier transform, used in the area of harmonic analysis

In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete Fourier transform.

Together with the Fourier number, the Biot number can be used in transient conduction problems in a lumped parameter solution which can be written as,