A note on constructing affine systems for L2

H. Q. Bui, N. Kaiblinger, R. S. Laugesen

Research output: Contribution to journalLetterpeer-review


Assume ψ ∈ L2 (Rd) has Fourier transform continuous at the origin, with over(ψ, ̂) (0) = 1, and that ∑l ∈ Zd | over(ψ, ̂) (ξ - l) |2 is bounded as a function of ξ ∈ Rd. Then every function f ∈ L2 (Rd) can be represented by an affine series f = ∑j > 0k ∈ Zd cj, k ψj, k for some coefficients satisfying{norm of matrix} c {norm of matrix}ℓ1 (ℓ2) = under(∑, j > 0) (under(∑, k ∈ Zd) | cj, k |2)1 / 2 < ∞ . Here ψj, k (x) = | det aj |1 / 2 ψ (aj x - k) and the dilation matrices aj expand, for example aj = 2j I. The result improves an observation by Daubechies that the linear combinations of the ψj, k are dense in L2 (Rd).

Original languageEnglish (US)
Pages (from-to)400-406
Number of pages7
JournalApplied and Computational Harmonic Analysis
Issue number3
StatePublished - Nov 2008


  • Completeness
  • Multiresolution analysis
  • Quasi-interpolation

ASJC Scopus subject areas

  • Applied Mathematics

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