# Quadratic Fourier transform
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In [mathematical physics](/source/mathematical_physics) and [harmonic analysis](/source/harmonic_analysis), the '''quadratic Fourier transform''' is an [integral transform](/source/integral_transform) that generalizes the [fractional Fourier transform](/source/fractional_Fourier_transform), which in turn generalizes the [Fourier transform](/source/Fourier_transform).[{{cite book|first1=Maurice A. de|last1=Gosson|title=Symplectic Methods in Harmonic Analysis and in Mathematical Physics|url=https://www.springer.com/gp/book/9783764399917|publisher=Birkhäuser Basel|date= 2011 |isbn=978-3-7643-9991-7|series=Pseudo-Differential Operators, Virtual Series on Symplectic Geometry|via=www.springer.com}}]
Roughly speaking, the Fourier transform corresponds to a change of variables from time to frequency (in the context of harmonic analysis) or from position to momentum (in the context of [quantum mechanics](/source/quantum_mechanics)). In [phase space](/source/phase_space), this is a 90 degree rotation. The fractional Fourier transform generalizes this to any angle rotation, giving a smooth mixture of time and frequency, or of position and momentum. The quadratic Fourier transform extends this further to the group of all linear [symplectic transformations](/source/Symplectic_group) in phase space (of which rotations are a subgroup).
More specifically, for every member of the [metaplectic group](/source/metaplectic_group) (which is a double cover of the symplectic group) there is a corresponding quadratic Fourier transform.