Sobolev Spaces and Approximation by Affine Spanning Systems

The dilations {aj} are lacunary, for example aj = 2 , and the coefficients cj,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(Rd) the scale averaging is unnecessary and one has the simpler formula f(x) = limj→∞ P k∈Zd cj,kψ(ajx − 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(ajx− k) : j > 0, k ∈ Z} in W (R). 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?