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Laplace Transform in the context of Gelfand/Pontryagin
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Is there a class of objects (presumably related to locally compact abelian groups) for which the quasi-characters canonically generalize the Laplace transform?
If not, is there a ...
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Where does the Laplace transform come from?
The Gelfand transform on the commutative Banach *-algebra $L^1(\mathbb{R})$ is just the Fourier transform.
Q. What can we say concerning the Laplace transform?
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