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Fourier and optics
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.
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.

Fourier and is
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.

Fourier and study
The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc.
In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.
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 propagation.
In mathematics, the term Fourier analysis often refers to the study of both operations.
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms.
The motivation for the Fourier transform comes from the study of Fourier series.
In the study of Fourier series, complicated but periodic functions are written as the sum of simple waves mathematically represented by sines and cosines.
In the study of Fourier series the numbers c < sub > n </ sub > could be thought of as the " amount " of the wave present in the Fourier series of ƒ.
The study of Fourier series is a branch of Fourier analysis.
The Fourier series is named in honour of Joseph Fourier ( 1768 – 1830 ), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d ' Alembert, and Daniel Bernoulli.
Fourier used it as an analytical tool in the study of waves and heat flow.
After him, several authors applied BV functions to study Fourier series in several variables, geometric measure theory, calculus of variations, and mathematical physics.
Fourier series have a surprisingly diverse array of applications in many scientific fields, in particular in all of the phenomena involving seasonal periodicities mentioned above, and in wave motion, and hence in the study of radiation, of acoustics, of seismology, of modulation of radio waves in electronics, and of electric power engineering.

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