Affine Systems that Span Lebesgue Spaces

We establish rather weak conditions on under which the small scale affine system spans . The conditions are that the periodization of |ψ| be locally in L^p, that , and that the dilation matrices a_j are expanding, meaning . The periodization of ψ need not be constant; that is, the integer translates need not form a partition of unity. The proof involves explicitly approximating an arbitrary function f using a linear combination of the , with the coefficients in the linear combination being local average values of f .