  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
recovering the thermal functions at real times from the corresponding functions
at imaginary times, introduced as primary objects in the Matsubara formalism.
The key property on which we rely is the fact that the Fourier transforms 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 Matsubara
imaginary energies, respectively in the upper and lower half-planes. Starting
from the Fourier coefficients regarded as `data set', we then develop a method
based on the Pollaczek polynomials for constructing explicitly their analytic
interpolations.
