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In particular, it can be seen that the scaling property of the Fourier transform says that if we compress a function to x, its Fourier transform extends to ξ. The Fourier transform can be generalized to any locally compact abelian group. It is not possible to arbitrarily condense both a function and its Fourier transform. Convergence still holds if ER is taken as a cube with side length R. As an alternative to the Fourier transform, time-frequency analysis uses time-frequency transforms or time-frequency distributions to represent signals in a form that contains some time information and some frequency information - according to the uncertainty principle there is a trade-off between them. Equality is obtained for Gaussian as in the previous case. When the function is integrable, the Fourier transform is still uniformly continuous and the Riemann-Lebesgue lemma holds. In this case, the Riemann-Lebesgue lemma is valid; f̂(ξ) is a function on Ĝ that is zero at infinity. The Fourier transform is also a special case of the Gelfand Ebay Scraper (Read Much more) transform. The trade-off between compression of a function and its Fourier transform can be formalized in uncertainty principle form by viewing a function and its Fourier transform as conjugate variables with respect to the symplectic form in the time-frequency domain: From the point of view of the linear canonical transform, the Fourier transform is rotated 90° in the time-frequency domain and its symplectic form protects.
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