# wavelet

## Motivation

Wavelets^{} can be used to analyze functions^{} in ${L}_{2}(\mathbb{R})$ (the space of all Lebesgue absolutely square integrable functions defined on the real numbers to the complex numbers^{}) in much the same way the complex exponentials^{} are used in the Fourier transform^{}, but wavelets offer the advantage of not only describing the frequency content of a function, but also providing information on the time localization of that frequency content.

## Definition

A (more properly, an *orthonormal dyadic*) *wavelet* is a function $\psi (t)\in {L}_{2}(\mathbb{R})$ such that the family of functions

$${\psi}_{jk}\equiv {2}^{j/2}\psi ({2}^{j}t-k),$$ |

where $j,k\in \mathbb{Z}$, is an orthonormal basis in the Hilbert space^{} ${L}_{2}(\mathbb{R}).$

## Notes

The scaling factor of ${2}^{j/2}$ ensures that $\parallel {\psi}_{jk}\parallel =\parallel \psi \parallel =1$. These type of wavelets (the most popular), are known as dyadic wavelets because the scaling factor is a power of 2. It is not obvious from the definition that wavelets even exist, or how to construct one; the Haar wavelet is the standard example of a wavelet, and one technique used to construct wavelets. Generally, wavelets are constructed from a multiresolution analysis, but they can also be generated using wavelet sets.

Title | wavelet |

Canonical name | Wavelet |

Date of creation | 2013-03-22 14:26:41 |

Last modified on | 2013-03-22 14:26:41 |

Owner | swiftset (1337) |

Last modified by | swiftset (1337) |

Numerical id | 11 |

Author | swiftset (1337) |

Entry type | Definition |

Classification | msc 65T60 |

Classification | msc 46C99 |

Related topic | FourierTransform |

Related topic | MultiresolutionAnalysis |

Related topic | WaveletSet2 |

Defines | wavelet |

Defines | orthonormal dyadic wavelet |