### Abstract

We establish rather weak conditions on ψ ∈ L ^{p}(ℝ ^{d}) under which the small scale affine system {ψ(a _{j}x-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 ∫ _{R}d 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 language | English (US) |
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Pages (from-to) | 533-556 |

Number of pages | 24 |

Journal | Journal of Fourier Analysis and Applications |

Volume | 11 |

Issue number | 5 |

DOIs | |

State | Published - 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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## Cite this

*Journal of Fourier Analysis and Applications*,

*11*(5), 533-556. https://doi.org/10.1007/s00041-005-4049-2