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Are there any algorithms that numerically compute the continuos Fourier transform of a given function f? I find plenty of implementations of the discrete Fourier transform, using FFT, but, if I´m not mistaken, DFT is not a discrete approximation of the continuous Fourier transform, but a different, although related, concept.

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    $\begingroup$ The GSL has oscillatory integration schemes that do the integration. You could do the integration with any quadrature of your choosing, really. $\endgroup$ – Cameron Williams Jul 14 '13 at 19:35
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The fast Fourier transform (FFT) is used to compute numerical approximations to continuous Fourier transforms (CFT). This is not apart from its application or correspondence to Discrete Fourier of course. A numerical approximation of the CFT requires evaluating a large number of integrals, each with a different integrand, since the values of this integral for a large range of the variable are needed. The FFT can be effectively applied to this problem. There are however cases where FFT in brotherhood with DFT are not accurate; e.g. DFT is periodical and spectrum aliasing may occur, other approximations are elaboreted on the spot such as here>>>

See further Here>>> Here>>>

Also cross reference Here>>>

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