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Let $f \colon [0,\infty) \to [0,\infty)$ be a continuous function and define its Laplace transform by $$\mathcal{L}f(\lambda) = \int_0^\infty f(t)e^{-\lambda t}\, dt, \quad \forall \lambda > 0.$$ Are there results giving sufficient conditions on $\mathcal{L}f$ which guarantee that $f$ decays exponentially fast at infinity? I am mainly looking for conditions on the regularity and/or decay of the Laplace transform. I am not at all familiar with this subject so any reference is greatly appreciated.

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