Affine synthesis onto lebesgue and hardy spaces

H. Q. Bui, R. S. Laugesen

Research output: Contribution to journalArticlepeer-review


The affine synthesis operator Sc = Σj>0 Σk∈ℤd C j,k Ψ j,k is shown to map the mixed-norm sequence space ℓ1(ℓp) surjectively onto Lp(Kd) under mild conditions on the synthesizer ψ ∈ Lp(ℝd), 1 ≤ p < ∞, with ∫ℝd ψ dx = 1. Here ψ j,k(x) = |det a j| 1/p ψ(ajx - k), and the dilation matrices ft/ expand, for example aj = 2 jI. Affine synthesis further maps a discrete mixed Hardy space ℓ1 (h1) onto H1(ℝd). Therefore the H1-norm of a function is equivalent to the infimum of the norms of the sequences representing the function in the affine system: (Equation Presented) where z = {zℓ} is a discrete Riesz kernel sequence.

Original languageEnglish (US)
Pages (from-to)2203-2233
Number of pages31
JournalIndiana University Mathematics Journal
Issue number5
StatePublished - 2008


  • Analysis
  • Quasi-interpolation
  • Scale averaging
  • Spanning
  • Synthesis

ASJC Scopus subject areas

  • General Mathematics


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