It is well-known that,
if $f$ and $g$ are compactly supported continuous functions, then their convolution exists, and is also compactly supported and continuous (Hörmander 1983, Chapter 1).
Next,
Suppose $f\in L^{1}(\mathbb R)$ is given.
My Question is:
Can we expect to choose $\phi \in C_{c}^{\infty}(\mathbb R)$ with $\int_{\mathbb R}\phi(t)dt=1$ and the support of $f\ast \phi$ is contained in a compact set, that is, $\operatorname{supp} f\ast \phi \subset K;$ where $K$ is some compact set in $\mathbb R$ ?
Thanks,
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