42 Facts About Fourier transform

Fourier transform is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial frequency or temporal frequency.

The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.

Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function.

The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain.

Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.

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Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory.

Fourier transform can be generalized to functions of several variables on Euclidean space, sending a function of 'position space' to a function of momentum .

In general, functions to which Fourier transform methods are applicable are complex-valued, and possibly vector-valued.

The fast Fourier transform is an algorithm for computing the DFT.

The Fourier transform is denoted here by adding a circumflex to the symbol of the function.

The conventions chosen in this article are those of harmonic analysis, and are characterized as the unique conventions such that the Fourier transform is both unitary on and an algebra homomorphism from to, without renormalizing the Lebesgue measure.

For example, one uses the Stone–von Neumann theorem: the Fourier transform is the unique unitary intertwiner for the symplectic and Euclidean Schrodinger representations of the Heisenberg group.

In 1822, Fourier transform claimed that any function, whether continuous or discontinuous, can be expanded into a series of sines.

However, the Fourier transform is able to represent non-periodic waveforms as well.

Similarly, the Fourier transform represents the amplitude and phase of every sinusoid present in a function.

Fourier transform uses an integral that exploits properties of sine and cosine to recover the amplitude and phase of each sinusoid in a Fourier series.

Therefore, the Fourier transform goes from one space of functions to a different space of functions: functions which have a different domain of definition.

In other conventions, the Fourier transform has in the exponent instead of, and vice versa for the inversion formula.

On n L, this extension agrees with original Fourier transform defined on, thus enlarging the domain of the Fourier transform to + L .

Since the complete set of Hermite functions provides a resolution of the identity, the Fourier transform can be represented by such a sum of terms weighted by the above eigenvalues, and these sums can be explicitly summed.

Many of the standard properties of the Fourier transform are immediate consequences of this more general framework.

Fourier transform is related to the Laplace transform, which is used for the solution of differential equations and the analysis of filters.

Fourier transform can be defined in any arbitrary number of dimensions.

In other words, where is a Gaussian function with variance, centered at zero, and its Fourier transform is a Gaussian function with variance.

Fourier's original formulation of the transform did not use complex numbers, but rather sines and cosines.

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An absolutely integrable function for which Fourier transform inversion holds can be expanded in terms of genuine frequencies by.

The Fourier transform in is no longer given by an ordinary Lebesgue integral, although it can be computed by an improper integral, here meaning that for an function,.

Definition of the Fourier transform can be extended to functions in for by decomposing such functions into a fat tail part in plus a fat body part in.

The Fourier transform is an automorphism on the Schwartz space, as a topological vector space, and thus induces an automorphism on its dual, the space of tempered distributions.

Kaniadakis ?-Fourier transform is a ?-deformation of the Fourier transform, associated with the Kaniadakis statistics, which is defined as:.

Fourier transform can be defined for functions on a non-abelian group, provided that the group is compact.

The Fourier transform on compact groups is a major tool in representation theory and non-commutative harmonic analysis.

In signal processing terms, a function is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency resolution, but no time information: the magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by phase, and standing waves are not localized in time – a sine wave continues out to infinity, without decaying.

Fourier transform studied the heat equation, which in one dimension and in dimensionless units is.

The Fourier transform is used in magnetic resonance imaging and mass spectrometry.

Fourier transform is useful in quantum mechanics in two different ways.

Therefore, the Fourier transform can be used to pass from one way of representing the state of the particle, by a wave function of position, to another way of representing the state of the particle: by a wave function of momentum.

Fourier transform methods have been adapted to deal with non-trivial interactions.

Since the result of applying the Fourier transform is again a function, we can be interested in the value of this function evaluated at the value for its variable, and this is denoted either as f or as .

Sometimes used to express that the Fourier transform of a rectangular function is a sinc function, or.

Unlike the convention followed in this article, when the Fourier transform is defined this way, it is no longer a unitary transformation on .

Since the fundamental definition of a Fourier transform is an integral, functions that can be expressed as closed-form expressions are commonly computed by working the integral analytically to yield a closed-form expression in the Fourier transform conjugate variable as the result.