Affine synthesis onto L p when 0 < p≤ 1

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The affine synthesis operator Sc=∑j>0, ∑kℤd cj,kεj,k is shown to map the coefficient space ℓp (ℤ+×ℤ d ) surjectively onto L p (ℝ d ), for p (0,1]. Here ψ j,k (x)=|det∈a j |1/p ψ(a j x-k) for dilation matrices a j that expand, and the synthesizer ψ L p (ℝ d ) need satisfy only mild restrictions, for example, ψ L 1(ℝ d ) with nonzero integral or else with periodization that is real-valued, nontrivial and bounded below. An affine atomic decomposition of L p follows immediately: ∥∫∥ ∑ j>0,{kℤcj,k}|p{1/p}:f= ∑j>0,∑kdc j,kψj,k∼ Tools include an analysis operator that is nonlinear on L p .

Original languageEnglish (US)
Pages (from-to)235-266
Number of pages32
JournalJournal of Fourier Analysis and Applications
Issue number2
StatePublished - Apr 2008


  • Analysis
  • Nonlinear quasi-interpolation
  • Path connectedness
  • Riesz basis
  • Spanning
  • Synthesis

ASJC Scopus subject areas

  • Analysis
  • General Mathematics
  • Applied Mathematics


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