  Spherical needlets were introduced by Narcowich, Petrushev, and Ward to
provide a multiresolution sequence of polynomial approximations to functions on
the sphere. The needlet construction makes use of integration rules that are
exact for polynomials up to a given degree. The aim of the present paper is to
relax the exactness of the integration rules by replacing them with QMC designs
as introduced by Brauchart, Saff, Sloan, and Womersley (2014). Such integration
rules (generalised here by allowing non-equal cubature weights) provide the
same asymptotic order of convergence as exact rules for Sobolev spaces
$\mathbb{H}^s$, but are easier to obtain numerically. With such rules we
construct ``generalised needlets''. The paper provides an error analysis that
allows the replacement of the original needlets by generalised needlets, and
more generally, analyses a hybrid scheme in which the needlets for the lower
levels are of the traditional kind, whereas the new generalised needlets are
used for some number of higher levels. Numerical experiments complete the
paper.
