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Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a harmonic (or loop-based) decomposition.

Are there deep connections between these two transforms? The formulaic connection is clear, but is there something deeper?

(Maybe the answer will involve spectral theory?)

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    I'd love to see a more precise version of the answer. Some people are flagging it as "not an answer," but it seems like an incomplete and potentially interesting answer.2013-07-30

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I don't know what answer you are looking for but for example both Laplace and Fourier transform are a so called Gelfand Transform.

You can find good introduction to Gelfand Transform in nice book Functional analysis for probability and stochastic processes: an introduction, A. Bobrowski. Look into Chapter 6.

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    @xenom I think this is the kind of answer I'm looking for, but I'm going to wait to select an answer for a while just in case.2011-03-19
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Laplace transform and Fourier transform are both special cases of the http://en.wikipedia.org/wiki/Linear_canonical_transformation.

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    It seems the Laplace transform given by the LCT is the bilateral Laplace transform or two-sided Laplace transform (http://en.wikipedia.org/wiki/Two-sided_Laplace_transform).2015-02-05
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Fourier transform does not exist for every signal application.So by introducing the region of convergence in Fourier transform which is known as Laplace Transform one may have indirectly the Fourier transform of signal.