  At the heart of non-parametric spectral estimation, lies the dilemma known as
the bias-variance trade-off: low biased estimators tend to have high variance
and low variance estimators tend to have high bias. In 1982, Thomson introduced
a multitaper method where this trade-off is made explicit by choosing a target
bias resolution and obtaining a corresponding variance reduction. The method
became the standard in many applications. Its favorable bias-variance trade-off
is due to an empirical fact, conjectured by Thomson based on numerical
evidence: assuming bandwidth W and N time domain observations, the average of
the square of the first $K=\left\lfloor 2NW\right\rfloor$ 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 quantify the approximation error in
the L1 norm; the approximation error is then used to control the bias of the
multitaper estimator resulting from spectral leakage. This leads to new
performance bounds for the method, explicit in terms of the bandwidth W and the
number N of time domain observations. Our method is flexible and can be
extended to higher dimensions and different geometries.
