### Abstract

Assume ψ ∈ L^{2} (R^{d}) 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 ∈ R^{d}. Then every function f ∈ L^{2} (R^{d}) can be represented by an affine series f = ∑_{j > 0} ∑_{k ∈ Zd} c_{j, k} ψ_{j, k} for some coefficients satisfying{norm of matrix} c {norm of matrix}_{ℓ1 (ℓ2)} = under(∑, j > 0) (under(∑, k ∈ Z^{d}) | c_{j, k} |^{2})^{1 / 2} < ∞ . Here ψ_{j, k} (x) = | det a_{j} |^{1 / 2} ψ (a_{j} x - k) and the dilation matrices a_{j} expand, for example a_{j} = 2^{j} I. The result improves an observation by Daubechies that the linear combinations of the ψ_{j, k} are dense in L^{2} (R^{d}).

Original language | English (US) |
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Pages (from-to) | 400-406 |

Number of pages | 7 |

Journal | Applied and Computational Harmonic Analysis |

Volume | 25 |

Issue number | 3 |

DOIs | |

State | Published - Nov 1 2008 |

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### Keywords

- Completeness
- Multiresolution analysis
- Quasi-interpolation

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

^{2}.

*Applied and Computational Harmonic Analysis*,

*25*(3), 400-406. https://doi.org/10.1016/j.acha.2008.04.001