  Conventional algorithms for galaxy power spectrum estimation measure the true
spectrum convolved with a survey window function, which, for parameter
inference, must be compared with a similarly convolved theory model. In this
work, we directly estimate the unwindowed power spectrum multipoles using
quadratic estimators akin to those introduced in the late 1990s. Under Gaussian
assumptions, these are optimal and free from the leading-order effects of
pixellization and non-Poissonian shot-noise. They may be straightforwardly
computed given the survey data-set and a suite of simulations of known
cosmology. We implement the pixel-based maximum-likelihood estimator and a
simplification based on the FKP weighting scheme, both of which can be computed
via FFTs and conjugate gradient descent methods. Furthermore, the estimators
allow direct computation of spectrum coefficients in an arbitrary linear
compression scheme, without needing to first bin the statistic. Applying the
technique to a subset of the BOSS DR12 galaxies, we find that the pixel-based
quadratic estimators give statistically consistent power spectra, compressed
coefficients, and cosmological parameters to those obtained with the usual
windowed approaches. Due to the sample's low number density and compact window
function, the optimal weighting scheme gives little improvement over the
simplified form; this may change for dense surveys or those focusing on
primordial non-Gaussianity. The technique is shown to be efficient and robust,
and shows significant potential for measuring the windowless power spectrum and
bispectrum in the presence of weak non-Gaussianity.
