Multirésolution

Multirésolution

Définition

Une suite (V_{j})_{j\in \mathbb{Z}} de sous-espaces vectoriels fermés de L^2(\mathbb{R}) est une approximation multirésolution si elle vérifie les 6 propriétés suivantes :

1)\forall j\in\mathbb{Z}, f\in V_{j} \iff f(2\cdot)\in V_{j-1}

2)\forall j \in \mathbb{Z}, V_{j+1} \subset V_{j}

3)\forall (j,k)\in \mathbb{Z}^2, f(\cdot)\in V_{j} \iff f(\cdot-2^{j}k)\in V_{j}

4)\lim_{j \rightarrow + \infty} V_{j} = \bigcap_{j\in\mathbb{Z}}V_{j} =\{ 0 \}

5)\lim_{j \rightarrow - \infty} V_{j} = \overline{\bigcup_{j\in\mathbb{Z}}V_{j}}=L^2(\mathbb{R})

6) Il existe \theta\in V_{0} tel que (\theta_{n})_{n\in \mathbb{Z}} soit une base de Riesz de V0.


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