Measurement of the angular power spectrum of the cosmic microwave background is most
often based on a spherical harmonic analysis of the observed temperature anisotropies. Even
if all-sky maps are obtained, however, it is likely that the region around the Galactic plane will
have to be removed as a result of its strong microwave emissions. The spherical harmonics are
not orthogonal on the cut sky, but an orthonormal basis set can be constructed from a linear
combination of the original functions. Previous implementations of this technique, based on
Gram–Schmidt orthogonalization, were limited to maximum Legendre multipoles of
lmax & 50, as they required all the modes have appreciable support on the cut-sky, whereas for
large lmax the fraction of modes supported is equal to the fractional area of the region retained.
This problem is solved by using a singular value decomposition to remove the poorly
supported basis functions, although the treatment of the non-cosmological monopole and
dipole modes necessarily becomes more complicated. A further difficulty is posed by
computational limitations – orthogonalization for a general cut requires Oðl
6
maxÞ operations
and Oðl
4
maxÞ storage and so is impractical for lmax * 200 at present. These problems are
circumvented for the special case of constant (Galactic) latitude cuts, for which the storage
requirements scale as Oðl
2
maxÞ and the operations count scales as Oðl
4
maxÞ. Less clear, however,
is the stage of the data analysis at which the cut is best applied. As convolution is ill-defined
on the incomplete sphere, beam-deconvolution should not be performed after the cut and, if
all-sky component separation is as successful as simulations indicate, the Galactic plane
should probably be removed immediately prior to power spectrum estimation.