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 $(F_1,F_2)={\int }_0^{\infty }(F_1(t)F_2(t)+F_1^{\prime }(t)F_2^{\prime }(t))dt$ we shall establish a real inversion formula for the Laplace transform.