  The capacity of the linear time-varying (LTV) channel, a continuous-time LTV
filter with additive white Gaussian noise, 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. Constraints on the average energy or on the squared-error
distortion, respectively, are used. 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 Szego theorem for a class of operators
associated with the filter. A self-contained proof of the Szego theorem is
given. The waterfilling theorems compare well with the classical results of
Gallager and Berger. In the case of a nonstationary source, it is observed that
the part of the classical power spectral density is taken by the Wigner-Ville
spectrum. The present approach is based on the spread Weyl symbol of the LTV
filter, and is asymptotic in nature. For the spreading factor, a lower bound is
suggested by means of an uncertainty inequality.
