## Abstract

We show that every function in the Hardy space can be approximated by linear combinations of translates and dilates of a synthesizer ψ ∈ L^{1} (R^{d}), provided only that ψ(0)=1 and ψ satisfies a mild regularity condition. Explicitly, we prove scale averaged approximation for each f ∈ H^{1}(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 H^{1} in terms of differences of ψ.

Original language | English (US) |
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Pages (from-to) | 149-172 |

Number of pages | 24 |

Journal | Constructive Approximation |

Volume | 28 |

Issue number | 2 |

DOIs | |

State | Published - Aug 1 2008 |

## Keywords

- Atomic
- Completeness
- Quasi-interpolation
- Scale averaging

## ASJC Scopus subject areas

- Analysis
- Mathematics(all)
- Computational Mathematics