  We use operator product expansion (OPE) techniques to study the spectral
functions of currents and stress tensors at finite temperature, in the
high-energy time-like region $\omega\gg T$. The leading corrections to these
spectral functions are proportional to $\sim T^4$ expectation values in
general, and the leading corrections $\sim g^2T^4$ are calculated at weak
coupling, up to an undetermined coefficient in the shear viscosity channel.
Spectral functions are shown to be infrared safe, in the deeply virtual regime,
up to order $g^8T^4$. The convergence of (vacuum subtracted) sum rules in the
shear and bulk viscosity channels is established in QCD to all orders in
perturbation theory, though numerically significant tails $\sim
T^4/(\log\omega)^3$ are shown to exist in the bulk viscosity channel. We argue
that the spectral functions of currents and stress tensors in infinitely
coupled $\mathcal{N}=4$ super Yang-Mills do not receive any medium-dependent
power correction.
