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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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    $\begingroup$ In what sense does the Laplace transform give a power-series decomposition? I don't understand the relationship between this question and the question you linked to. $\endgroup$ Commented Mar 18, 2011 at 23:03
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    $\begingroup$ The obvious link is more natural and pertinent, I think, that the question you linked. en.wikipedia.org/wiki/Laplace_transform#Fourier_transform $\endgroup$ Commented Mar 19, 2011 at 0:00
  • $\begingroup$ @Qiaochu Yuan A power series says what constants $\vec{a}$ will make $\sum a_i x^i = f(x)$. The Laplace (Mellin) transform says what function $a(i)$ will make $\int a(i) x^i = f(x)$. In the linked Q, @Christian Blatter's answer gives $F(phi) = \sum_{k=0}^n a_k e^{i k \phi}$. $\endgroup$ Commented Mar 19, 2011 at 0:05
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    $\begingroup$ Laplace can only mutiply or divide the signals. Fourier can only add or subtract the signals $\endgroup$ Commented Jul 30, 2013 at 14:58
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    $\begingroup$ 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. $\endgroup$ Commented Jul 30, 2013 at 15:43

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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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  • $\begingroup$ @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. $\endgroup$ Commented Mar 19, 2011 at 0:13
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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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    $\begingroup$ This is an interesting answer but more detail would be nice. Particularly, the Laplace transform given by the LCT isn't quite the same as the standard textbook definition of the Laplace transform (integration range mismatch). $\endgroup$ Commented Feb 4, 2015 at 20:01
  • $\begingroup$ It seems the Laplace transform given by the LCT is the bilateral Laplace transform or two-sided Laplace transform (en.wikipedia.org/wiki/Two-sided_Laplace_transform). $\endgroup$ Commented Feb 5, 2015 at 9:44
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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.

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