Approximation and Spanning in the Hardy Space, by Affine Systems

Abstract

We show that every function in the Hardy space can be approximated by linear combinations of translates and dilates of a synthesizer \(\psi \in L^1({\bf R}^d)\), provided only that \(\widehat{\psi}(0)=1\) and \(\psi\) satisfies a mild regularity condition. Explicitly, we prove scale averaged approximation for each \(f \in H^1({\bf R}^d)\), $$f(x) = \lim_{J \to \infty} \frac{1}{J} \sum_{j=1}^J \sum_{k \in {\bf Z}^d} c_{j,k} \psi(a_j x - k),$$ 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 \(1<p<\infty\). A corollary is a new affine decomposition of \(H^1\) in terms of differences of \(\psi\).