The inverse Laplace transform and analytic pseudo-differential operators (Q1276364)

The authors use a spectral theoretic approach to express the inverse of the Laplace transform \(L\) in the form \[ L^{-1}= (1/\pi)V^{-1} \cos(\pi D)VL, \tag{1} \] where \(V\) is a unitary transform and \(D= \frac{d}{dx}\). For suitably restricted \(F\), it is shown that \[ \cos(\pi D)VLF(x)= \text{Re} [VLF(x+i\pi)].\tag{2} \] Applications of the relations (1) and (2) to specific examples are also considered.