A unitary linear operator which resolves a function on $\mathbb{R}^N$ into a linear superposition of "plane wave functions". Most often used in physics for calculating the response of a time shift invariant linear system as the sum of its response to time harmonic excitation or for transforming a quantum state in position co-ordinates into one in momentum co-ordinates and contrawise. There is also a discrete, fast Fourier transform for discretised data.

Mathematical background.

The Fourier transform of a function $f\in L^1(\mathbb R^n)$ is defined as the function $\tilde f\in L^\infty(\mathbb R^n)$ given by $$ \tilde f(k)\equiv \int_{\mathbb R^n} \mathrm e^{i\langle k,x\rangle}f(x)\ \mathrm dx $$ where $\langle \cdot,\cdot\rangle\colon\mathbb R^n\times\mathbb R^n\to\mathbb R$ is some scalar product, typically, $\langle k,x\rangle\equiv k_1x_1+\cdots,k_nx_n$.

By the Fourier inversion theorem, if $\tilde f\in L^1(\mathbb R)$ then we may recover $f$ from $\tilde f$, through $$ f(x)\equiv \frac{1}{(2\pi)^n}\int_{\mathbb R^n} \mathrm e^{-i\langle k,x\rangle}\tilde f(k)\ \mathrm dk\qquad (\text{a.e.}) $$ where equality is understood in the sense of $L^p$.

One should point out that there are several different conventions regarding Fourier transforms. For example, some people redefine $\tilde f\to (2\pi)^{n/2}\tilde f$ so that the factors of $2\pi$ appear symmetrically in the transform and its inverse. Similarly, some people use $\langle\cdot,\cdot\rangle\to 2\pi\langle \cdot,\cdot\rangle$ so as to eliminate all factors of $2\pi$ from the integration measures. Other conventions are summarised in the wikipedia page.

It should be mentioned that one may extend the definition of the Fourier transform to spaces other than $L^1(\mathbb R^n)$. For example, by continuity (and dominated convergence) it is easy to extend the Fourier transform to $L^2(\mathbb R^n)$. Similarly, by duality it is straightforward to define the Fourier transform of distributions (most importantly, tempered distributions, in which case the Fourier transform is an isomorphism).

Finally, one should point out that it is possible and convenient to understand Fourier transforms with respect to measures more general than the pair $\mathrm dx,\mathrm dk$. This is best understood by means of the so-called Pontryagin duality which relates the domain of $f$ to that of $\tilde f$. In this context, one may understand the Fourier series as a special case of the Fourier transform, where the measure is just the counting measure.

Physical background.

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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