  The natural approach to a spectral analysis of data distributed on the sky
employs spherical harmonic decomposition. A common problem encountered in
practical astronomy is the lack of full sky coverage in the available data. For
example, the removal of the galactic plane data from the COBE DMR sky maps
compromises Fourier analysis of the cosmic microwave background (CMB)
temperature distribution due to the loss of orthogonality of the spherical
harmonics. An explicit method for constructing orthonormal functions on an
incomplete (e.g. Galaxy-cut) sphere is presented. These functions should be
used in the proper Fourier analysis of the COBE DMR sky maps to provide the
correct input for the determination of the spectrum of primordial
inhomogeneity. The results of such an analysis are presented in an accompanying
Letter. A similar algebraic construction of appropriate functions can be
devised for other astronomical applications.
