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The Fourier transform of a function $f:\mathbb{R}^N\to \mathbb{C}$, when it exists, is also a function $\tilde{F}:\mathbb{R}^N\to \mathbb{C}$ and is defined as:

$$\tilde{F}(k_1, k_2, \cdots, k_N) = \frac{1}{(2 \pi)^\frac{N}{2}}\int_{-\infty}^\infty \cdots \int_{-\infty}^\infty \exp(i(k_1 x_1+\cdots + k_N x_N)) f(x_1,\cdots,x_n)\mathrm{d}x_1\cdots \mathrm{d}x_N$$

and its inverse is then:

$$ f(x_1,\cdots,x_n) = \frac{1}{(2 \pi)^\frac{N}{2}}\int_{-\infty}^\infty \cdots \int_{-\infty}^\infty \exp(-i(k_1 x_1+\cdots + k_N x_N)) \tilde{F}(k_1, \cdots, k_N)\mathrm{d}k_1\cdots \mathrm{d}k_N$$

These are the unitary versions - other, similar definitions are often used. The Fourier transform resolves a function into a plane wave or sinusoidal function superposition, which is useful for analysing the response of shift-invariant linear systems. For example, one can compute time-harmonic field solutions to Maxwell's equations, and then build up more general solutions. The Fourier transform is also the unitary transformation between position and momentum co-ordinates (i.e. the two co-ordinate systems wherein the position $\hat{\mathbf{x}}$ and $\hat{\mathbf{p}}$ observables are the simple multiplication operators $\hat{\mathbf{x}} g(x) = x g(x)$ and $\hat{\mathbf{p}} h(p) = p h(p)$) for any pair of observables $\hat{\mathbf{x}}$ and $\hat{\mathbf{p}}$ fulfilling the canonical commutation relationship $[\hat{\mathbf{x}}, \hat{\mathbf{p}}] = i\hbar$.

The natural domain of definition of the Fourier transform is the space of tempered distributions. Every tempered distribution has a Fourier transform which is a tempered distribution, and the Fourier transform's kernel on this space trivially contains only the zero distribution. Thus, a tempered distribution and its Fourier transform constitute precisely the same information.

There is a discrete version of the Fourier transform and its inverse used for evenly sampled discretised data, wontedly implemented by the Fast Fourier Transform.

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