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By exploiting the analyticity and boundary value properties of the thermalGreen functions that result from the KMS condition in both time and energycomplex variables, we treat the general (non-perturbative) problem ofrecovering the thermal functions at real times from the corresponding functionsat imaginary times, introduced as primary objects in the Matsubara formalism.The key property on which we rely is the fact that the Fourier transforms ofthe retarded and advanced functions in the energy variable have to be the`unique Carlsonian analytic interpolations' of the Fourier coefficients of theimaginary-time correlator, the latter being taken at the discrete Matsubaraimaginary energies, respectively in the upper and lower half-planes. Startingfrom the Fourier coefficients regarded as `data set', we then develop a methodbased on the Pollaczek polynomials for constructing explicitly their analyticinterpolations.Comment: 23 pages, 2 figure

Reconstructing the Thermal Green Functions¶at Real Times from Those at Imaginary Times