Approximation and spanning in the hardy space, by affine systems

We show that every function in the Hardy space can be approximated by linear combinations of translates and dilates of a synthesizer ψ ∈ L¹ (R^d), provided only that ψ(0)=1 and ψ satisfies a mild regularity condition. Explicitly, we prove scale averaged approximation for each f ∈ H¹(R^d), f(x) = lim _J→∞ 1/J ∑_j=1^J ∑_k∈Zd c_j,k ψ(a_j x - k), where a _j is an arbitrary lacunary sequence (such as a_j=2 ^j) and the coefficients c_j,k are local averages of f. This formula holds in particular if the synthesizer ψ is in the Schwartz class, or if it has compact support and belongs to L^p for some 1<p<∞. A corollary is a new affine decomposition of H¹ in terms of differences of ψ.