Ondelette de Haar

L'ondelette de Haar

L'ondelette de Haar, ou fonction de Rademacher, est une ondelette créée par Alfréd Haar en 1909[1]. On considère que c'est la première ondelette connue. Il s'agit d'une fonction constante par morceaux, ce qui en fait l'ondelette la plus simple à comprendre et à implémenter. L'ondelette de Haar peut être généralisée par ce qu'on appelle le système de Haar.

Ondelette de Haar

La fonction-mère des ondelettes de Haar est une fonction constante par morceaux :

La fonction d'échelle associée est alors une fonction porte :

Le système de Haar

Le système de Haar est une suite de fonctions continues par morceaux, appartenant à pour . Il est défini de la manière suivante, à partir des fonctions indicatrices :

  • Pour et  :

Voici les représentations graphiques de h2 et de h3 :

Une des propriétés intéressantes du système de Haar est qu'il est une base de Schauder de pour .

Références

Sur les autres projets Wikimedia :

  1. (en) « Wavelets: seeing the Forest - and the Trees », sur www.beyonddiscovery.org (consulté le )

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