  Thomson's multitaper method estimates the power spectrum of a signal from $N$
equally spaced samples by averaging $K$ tapered periodograms. Discrete prolate
spheroidal sequences (DPSS) are used as tapers since they provide excellent
protection against spectral leakage. Thomson's multitaper method is widely used
in applications, but most of the existing theory is qualitative or asymptotic.
Furthermore, many practitioners use a DPSS bandwidth $W$ and number of tapers
that are smaller than what the theory suggests is optimal because the
computational requirements increase with the number of tapers. We revisit
Thomson's multitaper method from a linear algebra perspective involving
subspace projections. This provides additional insight and helps us establish
nonasymptotic bounds on some statistical properties of the multitaper spectral
estimate, which are similar to existing asymptotic results. We show using
$K=2NW-O(\log(NW))$ tapers instead of the traditional $2NW-O(1)$ tapers better
protects against spectral leakage, especially when the power spectrum has a
high dynamic range. Our perspective also allows us to derive an
$\epsilon$-approximation to the multitaper spectral estimate which can be
evaluated on a grid of frequencies using $O(\log(NW)\log\tfrac{1}{\epsilon})$
FFTs instead of $K=O(NW)$ FFTs. This is useful in problems where many samples
are taken, and thus, using many tapers is desirable.
