Sobolev spaces and approximation by affine spanning systems

H. Q. Bui, R. S. Laugesen

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We develop conditions on a Sobolev function ψ ∈ W m,p}(ℝd) such that if ψ̂(0) = 1 and ψ satisfies the Strang-Fix conditions to order m - 1, then a scale averaged approximation formula holds for all f ∈ Wm,p} (ℝd) : f(x) = limJ→ ∞ 1/J ∑Jj=1k ∈ ℤd cj,kψ(aj x - k) in Wm, p (ℝd). The dilations { a j } are lacunary, for example a j = 2 j , and the coefficients c j,k are explicit local averages of f, or even pointwise sampled values, when f has some smoothness. For convergence just in Wm - 1,p(ℝd) the scale averaging is unnecessary and one has the simpler formula f(x) = lim j → ∞ k ∈ ℤd cj,kψ(aj x - k). The Strang-Fix rates of approximation are recovered. As a corollary of the scale averaged formula, we deduce new density or "spanning" criteria for the small scale affine system {ψ(aj x - k) : j > 0, k ∈ ℤd} in Wm,p(ℝd). We also span Sobolev space by derivatives and differences of affine systems, and we raise an open problem: does the Gaussian affine system span Sobolev space?

Original languageEnglish (US)
Pages (from-to)347-389
Number of pages43
JournalMathematische Annalen
Issue number2
StatePublished - Jun 2008

ASJC Scopus subject areas

  • General Mathematics


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