Let $f \in L^1(\mathbb{R})$ such that $f$ vanishes on a positive measure, compact set $K$ such that $K\subset [-a,a]$. Does there exist $a>b>0$ and $\phi\in C_c^∞(\mathbb{R})$ supported on $[-b,b]$ such that $f*\phi$ vanishes on a positive measure set?
Note: If $f$ vanishes on $[-a,a]$ and $\phi$ supported on $ [-b,b] $ where $b<a$ then $f*\phi $ vanishes on $[-a+b,a-b]$