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For the Fourier transform, there are various theorems formalizing a correspondence between the smoothness of a function and the rate of decay of its Fourier transform. For example, if a function is $n$ times continuously differentiable, then its Fourier transform asymptotically decays like a reciprocal $n$th power, something like this.

For the unilateral Laplace transform, are there corresponding results? In particular, are there natural conditions on a function which entail any desired particular decay rate on its unilateral Laplace transform?

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    $\begingroup$ The decay rate of Laplace transform is related to the behavior of function near 0, but not to smoothness at other points. For example, vanishing of $f$ to order $a$ at $0$ implies $|Lf(s)|<Cs^{-a-1},\; s\to+\infty$. Vanishing of $f$ on an interval near zero is equivalent to exponential decay of the Laplace transform. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 9, 2020 at 13:04
  • $\begingroup$ Oh, that's a great criterion, thanks! Where can I find a reference with more details on this? $\endgroup$
    – Sridhar Ramesh
    Commented Sep 9, 2020 at 17:14
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    $\begingroup$ Any book which treats Laplace transform. For example Folland, Fourier Analysis. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 9, 2020 at 22:06
  • $\begingroup$ Thanks! I've found the result there. I realize this was a very basic question to ask on MathOverflow, but I didn't know the answer, and now I do! $\endgroup$
    – Sridhar Ramesh
    Commented Sep 10, 2020 at 0:17

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