Orthogonalisation of scalar basis functions on an incomplete sky

Measurement of the angular power spectrum of the cosmic microwave background requires a spherical harmonic analysis of the observed temperature anisotropies. Even if all-sky maps are obtained, the region around the Galactic plane must be removed due to its strong microwave emissions, and the spherical harmonics are no longer orthonormal on the cut sky. An orthonormal basis set can be constructed from a linear combination of the spherical harmonics, but previous implementations of this technique were limited to maximum Legendre multipoles of lmax <∼ 100 due to numerical singularity of the covariance matrix and computational limitations – the general problem requires O(l max) operations and O(l 4 max) storage. The first problem is solved here by using singular-value decomposition to remove poorly-supported basis functions, and the computational restrictions are circumvented for the special case of constant (Galactic) latitude cuts; the method presented here requires only O(l max) operations and O(l max ) storage. However, this improved efficiency is only useful in conjunction with fast integration of products of the spherical harmonics, new recursion formulæ for which are derived here.