Products of Daubechies Operators

  • M. W. Wong
Part of the Operator Theory: Advances and Applications book series (OT, volume 136)


Let FL 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
$$\begin{array}{*{20}{c}} {\varphi (x) = {{\pi }^{{ - \tfrac{n}{4}}}}{{e}^{{ - \tfrac{{|x{{|}^{2}}}}{2}}}},} & {x \in {{\mathbb{R}}^{n}}.} \\ \end{array}$$
The starting point is the following theorem.


Real Number Positive Constant Linear Operator Characteristic Function Measurable Function 
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Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • M. W. Wong
    • 1
  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

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