### Abstract

Given a subgroup S of GL (n), let G be the semidirect product of S with ℝ^{n}. The wavelet transform is defined for functions in L^{2} (ℝ^{n}) by using the action of G on this space. The standard properties of the wavelet transform and its inverse are quickly and easily derived in this formalism. In particular, the admissibility condition for the wavelet is expressed in terms of an integral over S. The notion of orthogonal wavelet channels is defined, and the wavelet transform is decomposed in terms of them. Other operators on L^{2} (ℝ^{n}) can also be analyzed in terms of their mixing of wavelet channels. For n = 2 and n = 3, details are given for the expansion of an arbitrary wavelet transform in terms of angular wavelet channels. An example is provided for n = 2. The correspondence between angular channels and the spherical harmonic decomposition of the Fourier transform of the wavelet transform is also outlined.

Original language | English (US) |
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Pages (from-to) | 80-102 |

Number of pages | 23 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 32 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2000 |

### Keywords

- Channel models
- Orthogonal functions
- Signal reconstruction
- Spherical harmonics
- Unitary representations of locally compact groups
- Wavelet transform

### ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Applied Mathematics