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 psi is an element of L(1) (R(d)), provided only that (psi) over cap (0)=1 and psi satisfies a mild regularity condition. Explicitly, we prove scale averaged approximation for each f is an element of H(1) (R(d)), f(x)=lim(J ->infinity) 1/J (J)Sigma(j=1) Sigma(k is an element of Zd) c(j,k)psi(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 psi is in the Schwartz class, or if it has compact support and belongs to L(p) for some 1<p<infinity. A corollary is a new affine decomposition of H(1) in terms of differences of psi.