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Wavelet Multipliers

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

Abstract

Let σ ∈ L (ℝ n . Then we define the linear operator T σ : L 2(ℝ n L 2(ℝ n by
$$\begin{array}{*{20}{c}} {{{T}_{\sigma }}u = {{\mathcal{F}}^{{ - 1}}}\sigma \mathcal{F}u,} & {u \in {{L}^{2}}({{\mathbb{R}}^{n}}),} \\ \end{array}$$
where \(\mathcal{F}\) and \({{\mathcal{F}}^{{ - 1}}}\), sometimes denoted by û, of a function u in L 2(ℝ n is given by
$$\mathcal{F}u = \mathop{{\lim }}\limits_{{R \to \infty }} {{({{\chi }_{R}}u)}^{ \wedge }},$$
where XR is the characteristic function of the ball with center at the origin and radius R,
$$\begin{array}{*{20}{c}} {{{{({{\chi }_{R}}u)}}^{ \wedge }}(\xi ) = {{{(2\pi )}}^{{ - \tfrac{n}{2}}}}\int_{{{{\mathbb{R}}^{n}}}} {{{e}^{{ - ix \cdot \xi }}}} {{\chi }_{R}}(x)u(x)dx,} & {\xi \in {{\mathbb{R}}^{n}},} \\ \end{array}$$
and the convergence of (XRu) ^ to Fu is understood to be in L 2(ℝ n . It is a consequence of Plancherel’s theorem that T σ : L 2(ℝ n L 2(ℝ n is a bounded linear operator.

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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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