  A new integration method drastically improves the efficiency of the dark
matter direct detection calculation. In this work I introduce a complete,
orthogonal basis of spherical wavelet-harmonic functions, designed for the new
vector space integration method. This factorizes the numeric calculation into a
part that depends only on the astrophysical velocity distribution; a second
part, depending only on the detector form factor; and a scattering matrix
defined on the basis functions, which depends on the details of the dark matter
(DM) particle model (e.g. its mass). For common spin-independent DM-Standard
Model interactions, this scattering matrix can be evaluated analytically in the
wavelet-harmonic basis. This factorization is particularly helpful for the more
complicated analyses that have become necessary in recent years, especially
those involving anisotropic detector materials or more realistic models of the
local DM velocity distribution. With the new method, analyses studying large
numbers of detector orientations and DM particle models can be performed about
10 million times faster.
  This paper derives several analytic results for the spherical wavelets,
including an extrapolation in the space of wavelet coefficients, and a
generalization of the vector space method to a much broader class of linear
functional integrals. Both results are highly relevant outside the field of DM
direct detection.
