Wavelet Multipliers
M. W. Wong · Birkhäuser Basel eBooks · 2002
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.