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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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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. –  Qiaochu Yuan Mar 18 '11 at 23:03
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The obvious link is more natural and pertinent, I think, that the question you linked. en.wikipedia.org/wiki/Laplace_transform#Fourier_transform –  leonbloy Mar 19 '11 at 0:00
    
@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 $F(phi) = \sum_{k=0}^n a_k e^{i k \phi}$. –  isomorphismes Mar 19 '11 at 0:05
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Laplace can only mutiply or divide the signals. Fourier can only add or subtract the signals –  user88296 Jul 30 '13 at 14:58
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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. –  Thomas Andrews Jul 30 '13 at 15:43
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1 Answer

up vote 10 down vote accepted

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. –  isomorphismes Mar 19 '11 at 0:13
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