### Abstract

The affine synthesis operator Sc=∑_{j>0}, ∑_{kℤ}^{d} c_{j,k}ε_{j,k} is shown to map the coefficient space ℓ^{p} (ℤ_{+}×ℤ ^{d} ) surjectively onto L ^{p} (ℝ ^{d} ), for p (0,1]. Here ψ _{j,k} (x)=|det∈a _{j} |^{1/p} ψ(a _{j} x-k) for dilation matrices a _{j} that expand, and the synthesizer ψ L ^{p} (ℝ ^{d} ) need satisfy only mild restrictions, for example, ψ L ^{1}(ℝ ^{d} ) with nonzero integral or else with periodization that is real-valued, nontrivial and bounded below. An affine atomic decomposition of L ^{p} follows immediately: ∥∫∥ ∑ _{j>0,}∑_{{kℤ}c_{j,k}}|^{p}{1/p}:f= ∑_{j>0},∑_{k} ℤ ^{d}c _{j,k}ψ_{j,k}∼ Tools include an analysis operator that is nonlinear on L ^{p} .

Original language | English (US) |
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Pages (from-to) | 235-266 |

Number of pages | 32 |

Journal | Journal of Fourier Analysis and Applications |

Volume | 14 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2008 |

### Keywords

- Analysis
- Nonlinear quasi-interpolation
- Path connectedness
- Riesz basis
- Spanning
- Synthesis

### ASJC Scopus subject areas

- Analysis
- Mathematics(all)
- Applied Mathematics