Approximation and spanning in the hardy space, by affine systems

H. Q. Bui, R. S. Laugesen

Research output: Contribution to journalArticlepeer-review


We show that every function in the Hardy space can be approximated by linear combinations of translates and dilates of a synthesizer ψ ∈ L1 (Rd), provided only that ψ(0)=1 and ψ satisfies a mild regularity condition. Explicitly, we prove scale averaged approximation for each f ∈ H1(Rd), f(x) = lim J→∞ 1/J ∑j=1Jk∈Zd cj,k ψ(aj x - k), where a j is an arbitrary lacunary sequence (such as aj=2 j) and the coefficients cj,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 Lp for some 1<p<∞. A corollary is a new affine decomposition of H1 in terms of differences of ψ.

Original languageEnglish (US)
Pages (from-to)149-172
Number of pages24
JournalConstructive Approximation
Issue number2
StatePublished - Aug 1 2008


  • Atomic
  • Completeness
  • Quasi-interpolation
  • Scale averaging

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Computational Mathematics


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