Products of Daubechies Operators
Man Wah Wong · Birkhäuser Basel eBooks · 2002
Let F ∈ L 2(ℂ n and let ϕ ∈ L 2(ℝ n be such that $$\parallel \varphi {{\parallel }_{{{{L}^{2}}({{\mathbb{R}}^{n}})}}} = 1$$ . Then the Daubechies operator associated to the symbol F and the admissible wavelet (p is the bounded linear operator $${{D}_{{F,\varphi }}}:{{L}^{2}}({{\mathbb{R}}^{n}}) \to {{L}^{2}}({{\mathbb{R}}^{n}})$$ defined by (17.23) for all functions f and g in L 2(ℝ n . We give in this chapter a formula for the product of two Daubechies operators when the admissible wavelet ϕ is chosen to be the function given by 22.1 $$\begin{array}{*{20}{c}} {\varphi (x) = {{\pi }^{{ - \tfrac{n}{4}}}}{{e}^{{ - \tfrac{{|x{{|}^{2}}}}{2}}}},} & {x \in {{\mathbb{R}}^{n}}.} \\ \end{array}$$ The starting point is the following theorem.