Spanning and Sampling in Lebesgue and Sobolev Spaces

We establish conditions on ψ under which the small-scale affine system {ψ(ajx− k) : j ≥ J, k ∈ Zd} spans the Lebesgue space L(R) and the Sobolev space W(R), for 1 ≤ p < ∞ and J ∈ Z. The dilation matrices aj are expanding (meaning limj→∞ ‖a−1 j ‖ = 0) but they need not be diagonal. For spanning L our result assumes ∫ Rd ψ dx 6= 0 and, when p > 1, that the periodization of |ψ| or of 1{ψ 6=0} is bounded. But the periodization of ψ need not be constant; in other words, the functions {ψ(x − k) : k ∈ Zd} need not form a partition of unity like B-splines do. For spanning W we impose the Strang–Fix condition on ψ, but only to order m− 1 whereas earlier authors required order m. These spanning results follow from explicitly approximating an arbitrary function f by linear combinations of the ψ(ajx− k), with the coefficients being local averages of f .