We establish rather weak conditions on y Î L^p(R^d)\psi\in L^p(R^d) under which the small scale affine system $\{\psi(a_jx-k): j>0,k\in Z^d\}$\{\psi(a_jx-k): j>0,k\in Z^d\} spans L^p(R^d), 1 £ p < ¥L^p(R^d), 1\le p<\infty . The conditions are that the periodization of |ψ| be locally in L^p, that ò _R^dydx ¹ 0\int_{ R^d}\psi dx\not= 0 , and that the dilation matrices a_j are expanding, meaning ||a_j^-1||® 0 as j®¥\Vert a_j^{-1}\Vert\rightarrow 0 \textrm {as} j\rightarrow\infty . The periodization of ψ need not be constant; that is, the integer translates {y(x-k): k Î Z^d}\{\psi(x-k): k\in Z^d\} need not form a partition of unity. The proof involves explicitly approximating an arbitrary function f using a linear combination of the y(a_jx-k)\psi(a_jx-k) , with the coefficients in the linear combination being local average values of f .