  The rational covariance extension problem (RCEP) is an important problem in
systems and control occurring in such diverse fields as control, estimation,
system identification, and signal and image processing, leading to many
fundamental theoretical questions. In fact, this inverse problem is a key
component in many identification and signal processing techniques and plays a
fundamental role in prediction, analysis, and modeling of systems and signals.
It is well-known that the RCEP can be reformulated as a (truncated)
trigonometric moment problem subject to a rationality condition. In this paper
we consider the more general multidimensional trigonometric moment problem with
a similar rationality constraint. This generalization creates many interesting
new mathematical questions and also provides new insights into the original
one-dimensional problem. A key concept in this approach is the complete smooth
parametrization of all solutions, allowing solutions to be tuned to satisfy
additional design specifications without violating the complexity constraints.
As an illustration of the potential of this approach we apply our results to
multidimensional spectral estimation and image compression. This is just a
first step in this direction, and we expect that more elaborate tuning
strategies will enhance our procedures in the future.
