  We obtain estimates for the Mean Squared Error (MSE) for the multitaper
spectral estimator and certain compressive acquisition methods for multi-band
signals. We confirm a fact discovered by Thomson [Spectrum estimation and
harmonic analysis, Proc. IEEE, 1982]: assuming bandwidth $W$ and $N$ time
domain observations, the average of the square of the first $K=2NW$ Slepian
functions approaches, as $K$ grows, an ideal band-pass kernel for the interval
$[-W,W]$. We provide an analytic proof of this fact and measure the
corresponding rate of convergence in the $L^{1}$ norm. This validates a
heuristic approximation used to control the MSE of the multitaper estimator.
The estimates have also consequences for the method of compressive acquisition
of multi-band signals introduced by Davenport and Wakin, giving MSE
approximation bounds for the dictionary formed by modulation of the critical
number of prolates.
