A real inversion formula for the Laplace transform in a Sobolev space (Q1969395)

Let \(S\) be the Hilbert space of the real-valued absolutely continuous functions \(F\) on \([0,\infty)\) with \(F(0)=0\) and the inner product \[ (F_1,F_2) =\int^\infty_0 \bigl(F_1(t) F_2(t)+ F_1'(t) F_2'(t)\bigr)dt. \] For the Laplace transform \(f\) of \(F\in S\) a real inversion formula is established, which converges uniformly on \([0,\infty)\). The proofs are based on the reproducing kernel \((1/2)(e^{-|t-\tau|}- e^{-t-\tau})\) for \(S\) and on the identity \[ \|F\|^2= \sum^\infty_{n=0} {1\over n!(n+1)!} \int^\infty_0 \Bigl(\bigl(D_n f(x)\bigr)^2+ \biggl(D_n\bigl( xf(x)\bigr) \biggr)^2 \Bigr)dx \] with \(D_n= x^n\partial^n_x x\partial_x\).