Affine systems that span lebesgue spaces

H. Q. Bui, R. S. Laugesen

Research output: Contribution to journalArticle

Abstract

We establish rather weak conditions on ψ ∈ L p(ℝ d) under which the small scale affine system {ψ(a jx-k): j>0,k ∈ ℤ d} spans L p(R d), 1 ≤ p < ∞. The conditions are that the periodization of |ψ| be locally in L p, that ∫ Rd ψ dx ≠ 0, and that the dilation matrices a j are expanding, meaning ∥a -1 j∥ → 0 as j → ∞. The periodization of ψ need not be constant; that is, the integer translates {ψ (x-k): k ∈ ℤ d} need not form a partition of unity. The proof involves explicitly approximating an arbitrary function f using a linear combination of the ψ(a x-k j), with the coefficients in the linear combination being local average values of f.

Original languageEnglish (US)
Pages (from-to)533-556
Number of pages24
JournalJournal of Fourier Analysis and Applications
Volume11
Issue number5
DOIs
StatePublished - Oct 1 2005

Keywords

  • Approximate
  • Completeness
  • Density
  • Quasi-interpolation
  • Spanning
  • Strang-Fix

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Applied Mathematics

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