  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 due to
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 orthogonalisation, were limited to maximum
Legendre multipoles of l_max < 50 as they required all the modes have
appreciable support on the cut sky, whereas for large l_max 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 --
orthogonalisation for a general cut requires O(l_max^6) operations and
O(l_max^4) storage and so is impractical for l_max > 200 at present. These
problems are circumvented for the special case of constant (Galactic) latitude
cuts, for which the storage requirements scale as O(l_max) and the operations
count scales as O(l_max^4). 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.
