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

MSE estimates for multitaper spectral estimation and off-grid compressive sensing