## Abstract

We develop conditions on a Sobolev function ψ ∈ W ^{m,p}}(ℝ^{d}) 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 ∈ W^{m,p}} (ℝ^{d}) : f(x) = lim_{J}→ ∞ 1/J ∑^{J}_{j=1} ∑ _{k} ∈ ℤ^{d} c_{j,k}ψ(a_{j} x - k) in W^{m, p} (ℝ^{d}). 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 W^{m - 1,p}(ℝ^{d}) the scale averaging is unnecessary and one has the simpler formula f(x) = lim j → ∞ k ∈ ℤ^{d} c_{j,k}ψ(a_{j} 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(a_{j} x - k) : j > 0, k ∈ ℤ^{d}} in W^{m,p}(ℝ^{d}). 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 language | English (US) |
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Pages (from-to) | 347-389 |

Number of pages | 43 |

Journal | Mathematische Annalen |

Volume | 341 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2008 |

## ASJC Scopus subject areas

- General Mathematics