Sobolev spaces and approximation by affine spanning systems

We develop conditions on a Sobolev function ψ ∈ W ^m,p}(ℝ^d) such that if ψ̂(0) = 1 and ψ satisfies the Strang-Fix conditions to order m - 1, then a scale averaged approximation formula holds for all f ∈ W^m,p} (ℝ^d) : f(x) = lim_J→ ∞ 1/J ∑^J_j=1 ∑ _k ∈ ℤ^d c_j,kψ(a_j x - k) in W^{m, p} (ℝ^d). The dilations { a _j } are lacunary, for example a _j = 2 ^j , and the coefficients c _j,k are explicit local averages of f, or even pointwise sampled values, when f has some smoothness. For convergence just in W^{m - 1,p}(ℝ^d) the scale averaging is unnecessary and one has the simpler formula f(x) = lim j → ∞ k ∈ ℤ^d c_j,kψ(a_j x - 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 {ψ(a_j x - k) : j > 0, k ∈ ℤ^d} in W^m,p(ℝ^d). 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?