2
$\begingroup$

What are the necessary and/or sufficient conditions for a function holomorphic on a right half-plane to be a unilateral Laplace transform of ANY function, square integrable or not, for which the transform is defined? (As an improper Riemannian or Lebsegian integral or whatever)

If I'm correct, a function analytic on a right half-plane is the unilateral Laplace transform of an L2 function if and only if it's square integrable on every vertical strip in which it is analytic, with a finit limit superior as Re (s) -> +inf When I can safely assume that a function is a Laplace transform of other function? (check the comment sections) But the inverse Laplace transform of (1/root (s), which is NOT square integrable) has a well-defined Laplace transform- 1/root (s)

Improve this question
$\endgroup$
3
  • $\begingroup$ Which theorem? Paley-Wiener? $1/\sqrt{s}$ is not square integrable on any vertical line, and it is the Laplace transform of $1/\sqrt{\pi t}$ which is also not square integrable. So? What exactly is the question? $\endgroup$
    – Robert Israel
    Commented Jul 21, 2017 at 19:49
  • $\begingroup$ I mean, what are the necessary and/or sufficient conditions for a function holomorphic on a right half-plane to be a unilateral Laplace transform of ANY function, square integrable or not, for which the transform is defined? $\endgroup$
    – Just A Young Artist
    Commented Aug 4, 2017 at 8:38
  • $\begingroup$ It depends if you say that $pv.(\frac{1}{t-1})1_{t > 0}$ is a function, or if you mean $\lim_{\epsilon \to 0, n \to 0} \int_{[\epsilon,n]} f(t) e^{-st}d\mu$ $\endgroup$
    – reuns
    Commented Aug 4, 2017 at 23:08

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .