A Real Inversion Formula for the Laplace Transform in a Sobolev Space

Abstract

For the real-valued Sobolev-Hilbert space on $(0, \infty)$ comprising absolutely continuous functions $F = F(t)$ normalized by $F(0) = 0$ and equipped with the inner product $(F_1, F_2) = \int^{\infty}_0 (F_1(t)F_2(t) + F'_1(t)F'_2(t))dt$ we shall establish a real inversion formula for the Laplace transform.