Affine systems that span lebesgue spaces

We establish rather weak conditions on ψ ∈ L ^p(ℝ ^d) under which the small scale affine system {ψ(a _jx-k): j>0,k ∈ ℤ ^d} spans L ^p(R ^d), 1 ≤ p < ∞. The conditions are that the periodization of |ψ| be locally in L ^p, that ∫ _Rd ψ dx ≠ 0, and that the dilation matrices a _j are expanding, meaning ∥a ^-1 _j∥ → 0 as j → ∞. The periodization of ψ need not be constant; that is, the integer translates {ψ (x-k): k ∈ ℤ ^d} need not form a partition of unity. The proof involves explicitly approximating an arbitrary function f using a linear combination of the ψ(a ^x-k _j), with the coefficients in the linear combination being local average values of f.