### Abstract

A subgroup D of GL (n, <) is said to be admissible if the semidirect product of D and ℝ ^{n}, considered as a subgroup of the affine group on ℝ ^{n}, admits wavelets ψ ε L ^{2}(ℝ ^{n} ) satisfying a generalization of the Calderón reproducing, formula. This article provides a nearly complete characterization of the admissible subgroups D. More precisely, if D is admissible, then the stability subgroup D _{x} for the transpose action of D on ℝ ^{n} must be compact for a. e. x ε ℝ ^{n}; moreover, if Δ is the modular function of D, there must exist an a ε D such that |det a| ≠ Δ(a). Conversely, if the last condition holds and for a. e. x ε ℝ ^{n} there exists an ε > 0 for which the ε-stabilizer D _{x} ^{ε} is compact, then D is admissible. Numerous examples are given of both admissible and non-admissible groups.

Original language | English (US) |
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Pages (from-to) | 89-102 |

Number of pages | 14 |

Journal | Journal of Geometric Analysis |

Volume | 12 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2002 |

### Keywords

- admissable dialatin groups
- continuous wavelets
- discrete wavelets

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

*Journal of Geometric Analysis*,

*12*(1), 89-102. https://doi.org/10.1007/BF02930862