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In mathematical physics and harmonic analysis, the quadratic Fourier transform is an integral transform that generalizes the fractional Fourier transform, which in turn generalizes the Fourier transform.
Roughly speaking, the Fourier transform corresponds to a change of variables from time to frequency or from position to momentum. In 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 in phase space.
More specifically, for every member of the metaplectic group there is a corresponding quadratic Fourier transform.