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Let $(g_j)_{j\in \mathbb{N}} \subseteq L_1(\mathbb{R}^n)$ with $\Vert g_j\Vert_1 = 1$ and $g_j \geq 0$ for all $j\in\mathbb{N}$. Suppose that $\lim\limits_{j\to \infty } d_j = 0 $ where $d_j := \sup\{\Vert x \Vert : x \in \text{support of } g_j\}$. Show that for all $f \in L_1(\mathbb{R}^n)$ we have $$\lim\limits_{j\to\infty} \Vert f \ast g_j - f\Vert_1 = 0.$$

I think that what I have to do to solve this problem is to show that the convolution $ f \ast g_j $ becomes something similar to a convolution to a dirac's delta. However i have trouble using boundness of the supp

Mathz777
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  • You could mimic the standard proof, where $g_j$ is the standard smoothing/mollification kernel: first approximate $f$ be a continuous function with compact support. Then prove the claim for this continuous function using uniform continuity arguments. – daw Apr 25 '24 at 06:28
  • @daw Could you point to a proof of this style? – M.... Apr 25 '24 at 10:19
  • see eg this question https://math.stackexchange.com/questions/8504/are-the-smooth-functions-dense-in-either-mathcal-l-2-or-mathcal-l-1 – daw Apr 25 '24 at 12:29

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