Reconstructing the thermal Green functions at real times from those at imaginary times

By exploiting the analyticity and boundary value properties of the thermal Green functions that result from the KMS condition in both time and energy complex variables, we treat the general (non-perturbative) problem of recov ering the thermal functions at real times from the corresponding functions at imaginary times, introduced as primary objects in the Matsubara formalis m. The key property on which we rely is the fact that the Fourier transform s of the retarded and advanced functions in the energy variable have to be the "unique Carlsonian analytic interpolations" of the Fourier coefficients of the imaginary-time correlator, the latter being taken at the discrete M atsubara imaginary energies, respectively in the upper and lower half-plane s. Starting from the Fourier coefficients regarded as "data set", we then d evelop a method based on the Pollaczek polynomials for constructing explici tly their analytic interpolations.