\(\mathrm {Exercise \ \oplus \ Problem } \ 17 \)
\(\mathrm {Exercise \ \oplus \ Problem } \ 17 \) |
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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 $$ \lim_{\delta \to 0 } \| f*K_\delta -f \| =0 .$$ where \( K_\delta \) is a family of good kernels.
\(\mathcal{P}roof. \)
$$ \begin{aligned} \left\|f * K_{\delta}-f\right\| &=\int_{\mathbb{R}^{n}}\left|f * K_{\delta}(x)-f(x)\right| d x \\ &=\int_{\mathbb{R}^{n}}\left|\int_{\mathbb{R}^{n}} f(x-y) K_{\delta}(y) d y-f(x)\right| d x \\ &=\int_{\mathbb{R}^{n}}\left|\int_{\mathbb{R}^{n}}(f(x-y)-f(x)) K_{\delta}(y) d y\right| d x \\ & \leq \int_{\mathbb{R}^{n}} \int_{\mathbb{R}^{n}}|f(x-y)-f(x)|\left|K_{\delta}(y)\right| d y d x \\ &=\int_{\mathbb{R}^n} \int_{\mathbb{R}^{n}}|f(x-y)-f(x)|\left|K_{\delta}(y)\right| d x d y \\ &=\int_{|y| \leq \ell} \int_{\mathbb{R}^{n}}|f(x-y)-f(x)|\left|K_{\delta}(y)\right| d x d y\\ & \ \ \quad\ \ \quad +\int_{|y|>\ell} \int_{\mathbb{R}^{n}} | f(x-y)-f(x)| \\ &=I_1+I_2. \end{aligned} $$ Then for \( I_1\), $$ \begin{aligned} I_1&= \int_{|y| \leq \ell}\left|K_{\delta}(y)\right| \int_{\mathbb{R}^{n}}|f(x-y)-f(x)| d x d y \\& = \int_{|y| \leq \ell}\left|K_{\delta}(y)\right| \| f(x-y)-f(x) \| _ 1 d x d y \end{aligned} $$ where \( \ell > 0\). By the continuity of the Lebesgue integral i.e. \( \forall \varepsilon >0 , \exists \zeta >0 ,\) when \( |y|<\zeta , \ \|f(x-y)-f(x) \| _1< \varepsilon.\) Hence $$ \ I_1\leq \varepsilon \int_{|y| \leq \ell}\left|K_{\delta}(y)\right| d y \leq M \varepsilon. $$ Then for \( I_2\), $$ \begin{aligned} I_2=& \int_{|y|>\ell} \int_{\mathbb{R}^{n}} | f(x-y)-f(x)| |K_\delta | dxdy\\ \leq & 2\|f\| _{L^1(\mathbb R^n )} \int_{|y| > \ell}\left|K_{\delta}(y)\right | \end{aligned}$$ By the properties of the good kernel, we have \(\forall \varepsilon >0, \exists \theta >0 \) s.t. when \( \delta <\theta , \int_{|y| > \ell}\left|K_{\delta}(y)\right| < \varepsilon. \) Then we obtain $$\| f*K_\delta -f \| <(M+ 2\| f \| ) \varepsilon. $$ Hence $$ \lim_{\delta \to 0 } \| f*K_\delta -f \| =0. $$