Sobolev spaces and approximation by affine spanning systems

Abstract

We develop conditions on a Sobolev function $\psi \in W^{m,p}({\mathbb{R}}^d)$ such that if $\widehat{\psi}(0) = 1$ and ψ satisfies the Strang–Fix conditions to order m − 1, then a scale averaged approximation formula holds for all $f \in W^{m,p}({\mathbb{R}}^d)$ :

$$ f(x) = \lim_{J \to \infty} \frac{1}{J} \sum_{j=1}^{J} \sum_{k \in {{\mathbb{Z}}}^d} c_{j,k}\psi(a_j x - k) \quad {\rm in} W^{m, p}({{\mathbb{R}}}^d).$$

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 ${W^{m - 1,p}({\mathbb{R}}^d)}$ the scale averaging is unnecessary and one has the simpler formula $f(x) = \lim_{j \to \infty} \sum_{k \in {\mathbb{Z}}^d} c_{j,k}\psi(a_j x - k)$ . 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 $\{\psi(a_j x - k) : j > 0, k \in {\mathbb{Z}}^d \}$ in $W^{m,p}({\mathbb{R}}^d)$ . 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?