Sobolev spaces and approximation by affine spanning systems.

We develop conditions on a Sobolev function ∈Wm,p(ℝd) such that if ̂ (0) = 1 and satisfies the Strang-Fix conditions to order m - 1, then a scale averaged approximation formula holds for all ∈ Wm,p(ℝd): [FORMULA] The dilations {j } are lacunary, for example j = 2j, and the coefficients Cj,k are explicit local averages of , or even pointwise sampled values, when has some smoothness. For convergence just in Wm-1,p(ℝd) the scale averaging is unnecessary and one has the simpler formula (x) = limj→∞ ∑k∈ℤd cj,k (jx-k). The Strang-Fix rates of approximation are recovered. As a corollary of the scale averaged formula, we deduce new density or "spanning" criteria for the small scale affine system { (jx-k) : j > 0, k ∈ ℤd } in Wm,p(ℝd). We also span Sobolev space by derivatives and differences of affine systems, and we raise an open problem: does the Gaussian affine system span Sobolev space