\(\mathrm {Exercise \ \oplus \ Problem } \ 3 \)
\(\mathrm {Exercise \ \oplus \ Problem } \ 3 \) |
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\( \qquad \)你好，这里是我的个人网站数学分析的每周一题栏目(数学分析每周一题，其中数学分析指的是数学中的分析学, 主要包括微积分，实分析，复分析) \(\qquad \ \)——————Alina Lagrange
Suppose that \(\varphi \in \mathcal{S}(\mathbb{R})\) and \(\int_{\mathbb{R}}|\varphi|^{2} d x=1 { } \), show that $$ \int_{\mathbb{R}}|x \varphi(x)|^{2} d x \int_{\mathbb{R}} \xi^{2}|\hat{\varphi}(\xi)|^{2} d \xi \geq \frac{1}{16 \pi^{2}} $$
\(\mathcal{P}roof. \)
As $$ \ \varphi \in \mathcal{S}(\mathbb{R}) \Rightarrow \int_{\mathbb{R}}|\varphi(x)|^{2} d x=- \int_{\mathbb{R}} x \frac{d}{d x}|\varphi(x)|^{2} d x $$$$ =-\int_{\mathbb{R}} x \overline{\varphi(x)} \varphi^{\prime}(x)+x \varphi(x) \overline{\varphi^{\prime}(x)} d x \ $$ By Cauthy-Schwartz inequality and Plancherel formula and the Fourier transform $$ \begin{aligned} 1 &=\int_{\mathbb{R}}|\varphi(x)|^{2} d x \\ & \leq 2 \int_{\mathbb{R}}\left|x\|\varphi(x)\| \varphi^{\prime}(x)\right| d x \\ & \leq 2\|x \varphi(x)\|_{L^{2}(\mathbb{R})}\left\|\varphi^{\prime}(x)\right\|_{L^{2}(\mathbb{R})} \\ &=2\|x \varphi(x)\|_{L^{2}(\mathbb{R})}\left\|\widehat{\varphi^{\prime}(x)}\right\|_{L^{2}(\mathbb{R})} \\ &=2\|x \varphi(x)\|_{L^{2}(\mathbb{R})}\|2 \pi i \xi \hat{\varphi}(\xi)\|_{L^{2}(\mathbb{R})} \\ \quad \text { Note }:|\varphi|^{2}=\bar{\varphi} \varphi \end{aligned} $$ Hence we finished the proof.