I am recently studying properties about a good kernel, and came across a problem.
Definition: A kernel $K_\delta$ is 'good' if they are Lebesgue integrable and satisfy the following conditions for $\delta>0$:
- $\int_{\mathbb R^d}K_\delta(x)dx=1$
- $\int_{\mathbb R^d}|K_\delta(x)|dx\le A$
- For every $\eta>0$, $$\int_{|x|\ge\eta}|K_\delta(x)|dx\to0\text{ }\text{ }\text{ as }\delta\to0$$
Here $A$ is a constant independent of $\delta$
The question is to prove that for every point of continuity of an integrable function $f$, $(f\star K_\delta)(x)\to f(x)$ as $\delta \to 0$, where we define $$(f\star K_\delta)(x)=\int_{\mathbb R^d}f(x-y)K_\delta(y)dy$$