  Radio maps provide radio frequency metrics, such as the received signal
strength, at every location of a geographic area. These maps, which are
estimated using a set of measurements collected at multiple positions, find a
wide range of applications in wireless communications, including the prediction
of coverage holes, network planning, resource allocation, and path planning for
mobile robots. Although a vast number of estimators have been proposed, the
theoretical understanding of the radio map estimation (RME) problem has not
been addressed. The present work aims at filling this gap along two directions.
First, the complexity of the set of radio map functions is quantified by means
of lower and upper bounds on their spatial variability, which offers valuable
insight into the required spatial distribution of measurements and the
estimators that can be used. Second, the reconstruction error for power maps in
free space is upper bounded for three conventional spatial interpolators. The
proximity coefficient, which is a decreasing function of the distance from the
transmitters to the mapped region, is proposed to quantify the complexity of
the RME problem. Numerical experiments assess the tightness of the obtained
bounds and the validity of the main takeaways in complex environments.
