real analysis - When convolution of two functions has compact support?

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,