Laplace Transform: Are there theorems similar to the Bernstein Theorem? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T05:30:48Z http://mathoverflow.net/feeds/question/82391 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82391/laplace-transform-are-there-theorems-similar-to-the-bernstein-theorem Laplace Transform: Are there theorems similar to the Bernstein Theorem? florian 2011-12-01T15:57:17Z 2011-12-01T15:57:17Z <p>Bernstein's Theorem states, that if a function is completely monotonic, then it is the Laplace transform of an $L^1$-function. (E.g. Widder, "The Laplace Transform", Chapter IV, Theorem 19b)</p> <p>Are there any theorems of the same flavor, which characterize the inverse Laplace transform of functions, which are locally bounded or locally $L^p$?</p> <p>More precisely, for $$f(x)=\int_0^\infty dt \; e^{-xt} \; g(t) \; ,$$ what do I have to prove for $f$ to be able to conclude, that $g \in L_{\textrm{loc}}^p$, for some $1\le p &lt; \infty$?</p>