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.