\(\mathrm {Exercise \ \oplus \ Problem } \ 16 \)
\(\mathrm {Exercise \ \oplus \ Problem } \ 16 \) |
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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
Prove that if \( f\) is continuous, of moderate decrease, and \[\int_\mathbb R f(y) e^{-y^{2}} e^{2 x y} d y=0\] for all \(x \in \mathbb{R} \), then \( f=0\).
\(\mathcal{P}roof. \)
Let \(g(x)=e^{-x^{2}} \). Then \[\begin{aligned} (f * g)(x)& =\int_{\mathbb R } f(y) e^{-(x-y)^{2}} d y\\&=e^{-x^{2}} \int_\mathbb R f(y) e^{-y^{2}} e^{2 x y} d y\\&=0 . \end{aligned} \] for all \(x\). This implies that \[ \widehat{f * g}(\xi)=\hat{f}(\xi) \hat{g}(\xi)=\hat{f}(\xi) \sqrt{\pi} e^{-\pi^{2} \xi^{2}}=0 \] for all \( \xi \). So \( \hat{f}=0 \). And hence \(f=0 \).