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I am an undergraduate student writing an expository thesis on the complex-analytic proof of the Prime Number Theorem. I understand that applying the Mellin Transform to the partial sum of the van ...

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?

Questions: 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 ...