  The capacity of the heat channel, a linear time-varying (LTV) filter with
additive white Gaussian noise (AWGN), is characterized by waterfilling in the
time-frequency plane. Similarly, the rate distortion function for a related
nonstationary source is characterized by reverse waterfilling in the
time-frequency plane. The source is formed by the white Gaussian noise response
of the same LTV filter as before. The proofs of both waterfilling theorems rely
on a specific Szego theorem for a positive definite operator associated with
the filter. An essentially self-contained proof of the Szego theorem is given.
The waterfilling theorems compare well with classical results of Gallager and
Berger. In case of the nonstationary source it is observed that the part of the
classical power spectral density (PSD) is taken by the Wigner-Ville spectrum
(WVS).
