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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.

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Diagonalizable matrix tight-binding model

Given the following model for a particle on a lattice: $$ \hat{H} = -t \sum_j[\left|\ j-2\right\rangle \left\langle j \right|+ \left|j+2\right\rangle \left\langle j \right|]$$ We introduce vectors $\ …
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